Chain rule
•Download as PPTX, PDF•
0 likes•60 views
The chain rule deals with functions within functions and describes how to take the derivative. It states that if y = f(g(x)), then y' = f'(g(x)) * g'(x). Alternatively, it can be written as letting u = g(x), so y(u) and then taking dy/dx = dy/du * du/dx. Examples are worked out, such as finding the derivative of y = (2x + 5)2 and y = 5/(3x-5)3.
Report
Share
Report
Share
Recommended
Chain rule
The document discusses the chain rule for derivatives. The chain rule is used when taking the derivative of a function composed of an inner and outer function, such as y=f(g(x)). The chain rule states that the derivative of the outer function is multiplied by the derivative of the inner function: dy/dx = f’(g(x)) * g’(x). Some examples are worked out, such as finding the derivative of y=(2x+5)2 and y=sin(x7), to demonstrate how to apply the chain rule.
Sol73
The document presents two solutions for calculating the force (F) measured by a scale when a chain is dropped onto it.
The first solution models the chain as a single object, equating the net force on the chain to its change in momentum. This yields an equation for F equaling 3σ(L-y)g, where σ is the linear mass density, L is the total length, y is the height of the top of the chain, and g is acceleration due to gravity.
The second solution considers the normal force from the scale as having two components: the weight of chain on the scale, and the force to stop additional chain hitting the scale. Modeling the momentum change, this also results
Calc03 9
The document discusses derivatives of exponential and logarithmic functions. It shows that the derivative of an exponential function e^x is itself, e^x. It also shows that the derivative of the natural logarithm function ln(x) is 1/x using the definition of derivative and the chain rule. Finally, it provides the general formula for deriving logarithmic functions of any base as -1/(xln(b)).
Verlet's method
Verlet's method is a second order method for solving equations of motion that is more accurate than first order methods like Euler's method. It works by calculating the velocity at the midpoint of each time step using the acceleration from the beginning of the step. This time centering is what gives it second order accuracy. To calculate the final velocity and position, it uses the average of the velocities halfway between each step. Care must be taken to initialize the method by taking a half step back or forward in time to calculate the initial velocity.
Question on proof_of_convergence
This document defines several mathematical concepts related to limits and convergence, including balls, neighborhoods, open sets, closed sets, and convergence of sequences. It provides formal definitions for these terms and establishes the relationships between them. Key definitions include a neighborhood being a set containing at least one ball centered at a point, an open set being a set containing a ball around every point in the set, and a sequence converging to a limit if it eventually lies within any ball centered at that limit.
GodsMathProject
The document discusses the unusual property that the derivative of the function ex is itself. It proves this property through three methods: using the limit definition of a derivative, implicit differentiation, and applying the chain rule. A graph is also provided to visualize how at each x value, the slope of the function ex is equal to the y value. The document concludes that through these three rigorous methods and a supporting graph, it has been clearly proven that the derivative of ex is indeed itself.
Proof By Contradictions
This document discusses proof by contradiction in mathematics. It begins by defining proof by contradiction as proving the truth of a statement by showing that assuming the statement is false leads to a contradiction. The document then provides examples of proofs by contradiction, including:
1) Proving there is no greatest integer by supposing there is a greatest integer N and showing that N+1 would also be an integer, contradicting that N was the greatest.
2) Proving the square root of 2 is irrational by supposing it is rational and showing this leads to a contradiction.
3) Explaining the general steps in a proof by contradiction: assume the statement is false, show this assumption leads to a contradiction, and thus
ANALISIS RIIL 1 2.4 ROBERT G BARTLE
1) The document contains exercises from an introduction to real analysis course on the supremum property. It includes solutions to problems involving finding the supremum and infimum of sets, and proving properties about the supremum and infimum.
2) For problem 4, it is proven that if S is a nonempty bounded set, then the infimum of S is less than or equal to all elements of S, and the supremum of S is greater than or equal to all elements of S.
3) For problem 6, it is shown that if A and B are bounded nonempty subsets, then the supremum of A union B is the maximum of the supremums of A and B, and the inf
Recommended
Chain rule
The document discusses the chain rule for derivatives. The chain rule is used when taking the derivative of a function composed of an inner and outer function, such as y=f(g(x)). The chain rule states that the derivative of the outer function is multiplied by the derivative of the inner function: dy/dx = f’(g(x)) * g’(x). Some examples are worked out, such as finding the derivative of y=(2x+5)2 and y=sin(x7), to demonstrate how to apply the chain rule.
Sol73
The document presents two solutions for calculating the force (F) measured by a scale when a chain is dropped onto it.
The first solution models the chain as a single object, equating the net force on the chain to its change in momentum. This yields an equation for F equaling 3σ(L-y)g, where σ is the linear mass density, L is the total length, y is the height of the top of the chain, and g is acceleration due to gravity.
The second solution considers the normal force from the scale as having two components: the weight of chain on the scale, and the force to stop additional chain hitting the scale. Modeling the momentum change, this also results
Calc03 9
The document discusses derivatives of exponential and logarithmic functions. It shows that the derivative of an exponential function e^x is itself, e^x. It also shows that the derivative of the natural logarithm function ln(x) is 1/x using the definition of derivative and the chain rule. Finally, it provides the general formula for deriving logarithmic functions of any base as -1/(xln(b)).
Verlet's method
Verlet's method is a second order method for solving equations of motion that is more accurate than first order methods like Euler's method. It works by calculating the velocity at the midpoint of each time step using the acceleration from the beginning of the step. This time centering is what gives it second order accuracy. To calculate the final velocity and position, it uses the average of the velocities halfway between each step. Care must be taken to initialize the method by taking a half step back or forward in time to calculate the initial velocity.
Question on proof_of_convergence
This document defines several mathematical concepts related to limits and convergence, including balls, neighborhoods, open sets, closed sets, and convergence of sequences. It provides formal definitions for these terms and establishes the relationships between them. Key definitions include a neighborhood being a set containing at least one ball centered at a point, an open set being a set containing a ball around every point in the set, and a sequence converging to a limit if it eventually lies within any ball centered at that limit.
GodsMathProject
The document discusses the unusual property that the derivative of the function ex is itself. It proves this property through three methods: using the limit definition of a derivative, implicit differentiation, and applying the chain rule. A graph is also provided to visualize how at each x value, the slope of the function ex is equal to the y value. The document concludes that through these three rigorous methods and a supporting graph, it has been clearly proven that the derivative of ex is indeed itself.
Proof By Contradictions
This document discusses proof by contradiction in mathematics. It begins by defining proof by contradiction as proving the truth of a statement by showing that assuming the statement is false leads to a contradiction. The document then provides examples of proofs by contradiction, including:
1) Proving there is no greatest integer by supposing there is a greatest integer N and showing that N+1 would also be an integer, contradicting that N was the greatest.
2) Proving the square root of 2 is irrational by supposing it is rational and showing this leads to a contradiction.
3) Explaining the general steps in a proof by contradiction: assume the statement is false, show this assumption leads to a contradiction, and thus
ANALISIS RIIL 1 2.4 ROBERT G BARTLE
1) The document contains exercises from an introduction to real analysis course on the supremum property. It includes solutions to problems involving finding the supremum and infimum of sets, and proving properties about the supremum and infimum.
2) For problem 4, it is proven that if S is a nonempty bounded set, then the infimum of S is less than or equal to all elements of S, and the supremum of S is greater than or equal to all elements of S.
3) For problem 6, it is shown that if A and B are bounded nonempty subsets, then the supremum of A union B is the maximum of the supremums of A and B, and the inf
intro to Implicit differentiation
This document discusses implicit differentiation, which is a technique for taking the derivative of equations that cannot be solved explicitly for y as a function of x. It explains that when differentiating terms involving both x and y, the derivative of the y term is dy/dx. As an example, it shows the differentiation of xy using the product rule, which yields y + x*dy/dx. The document concludes by applying this technique to differentiate the equation y4 + xy = x3 - x + 2 implicitly with respect to x.
Composition and inverse of functions
The document discusses composition of functions and inverse functions. It defines composition of functions as combining two functions where one function is performed first and the result is substituted into the second function. The composition is not always commutative. It then provides examples of finding the composition of two functions. Inverse functions are defined as functions where the independent and dependent variables are swapped, and their composition is equal to x. The document demonstrates finding the inverse of functions by swapping variables and checking the composition. It emphasizes that the inverse of a function may not always be a function itself.
CHAIN RULE AND IMPLICIT FUNCTION
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
20 the chain rule
The document discusses general partial derivatives and the chain rule. It defines partial derivatives for functions of multiple variables as holding all other variables constant. It provides an example function z = f(v,w,x,y) and explains how to compute the partial derivatives dz/dx, dz/dy by treating the other variables as constants. The chain rule is introduced for functions of composite variables, where the derivative dz/dt is the sum of the products of partial derivatives along all paths from z to the variable t. An example using this chain rule is worked out in detail. Variable trees are presented as a way to visualize the relationships between composite variables in a chain.
_lecture_05 F_chain_rule.pdf
1. The document discusses the chain rule for functions of several variables. It provides examples of how to use a "tree diagram" to represent variable dependencies and derive the appropriate chain rule statement.
2. It also gives examples of applying the chain rule to find derivatives like df/dt for functions where variables like x, y, and z depend on t, or to find partial derivatives like ∂f/∂s and ∂f/∂t.
3. One example works through applying the chain rule to find df/dt for the function f(x,y) = x^2 + y^2, where x = t^2 and y = t^4, and verifies the
3.7 Inverse Functions
This document discusses inverse functions. It defines an inverse function as undoing another one-to-one function, where applying one function followed by the other results in the original input. The domain of the original function is the range of the inverse, and vice versa. To find the inverse function, interchange the variables of the original function and solve for the new dependent variable. Graphing the inverse results in a reflection of the original function across the line y=x, allowing it to check if the inverse was found correctly. Examples are provided to demonstrate finding and graphing inverse functions.
Partial Differentiation
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
Calculus
This document discusses the difference between ordinary derivatives and partial derivatives. Ordinary derivatives are taken with respect to one independent variable, while partial derivatives treat all other variables as constant except for the variable being differentiated. The document provides examples of calculating ordinary and partial derivatives for functions of two variables. It also notes that partial derivatives are used in the chain rule to compute ordinary derivatives for functions with multiple variables.
Limits and derivatives
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
Understanding the "Chain Rule" for Derivatives by Deriving Your Own Version
Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.
5.2 the substitution methods
The document discusses the substitution method of integration. It explains that while the derivative of an elementary function is another elementary function, the antiderivative may not be. There are two main integration methods: substitution and integration by parts. Substitution reverses the chain rule by letting u be a function of x with derivative u', then substituting u for x and replacing dx with du/u' in the integral.
Relations and functions
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
02 first order differential equations
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
The Chain Rule Powerpoint Lesson
a powerpoint calculus lesson on "The Chain Rule". This is the initial step of an ongoing lesson that we will add to as the class progresses.
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
The document discusses the chain rule for derivatives. It begins by defining function composition and provides examples of composing linear functions. It then states the chain rule theorem, which says that the derivative of a composition is the product of the individual function derivatives evaluated at the same point. Several examples are worked out applying the chain rule to find the derivative of various compositions of functions.
4.1 Inverse Functions
Review composition of functions and one-to-one functions
Identify inverse functions
Write the equation of an inverse function
Ch07
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
Calc 4.5a
This document discusses techniques for evaluating indefinite integrals using substitution, including pattern recognition, change of variables, and the general power rule. It explains that substitution helps evaluate more complex integrals that involve a composite function. Several examples demonstrate identifying the pattern f(g(x))g'(x) and performing u-substitutions to evaluate the integral. It also discusses the need to sometimes multiply and divide by a constant or perform a formal change of variables if the pattern does not exactly match.
Calc 4.5a
This document discusses techniques for integration using substitution. It explains that substitution allows for the integration of more complex composite functions by breaking them into an inside and outside part. The main techniques are pattern recognition, where the integrand fits the standard pattern of f(g(x))g'(x), and change of variables, where a substitution u is made to simplify the integrand. It provides examples of applying both pattern recognition and change of variables substitutions to evaluate definite and indefinite integrals.
Applications of partial differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
More Related Content
Similar to Chain rule
intro to Implicit differentiation
This document discusses implicit differentiation, which is a technique for taking the derivative of equations that cannot be solved explicitly for y as a function of x. It explains that when differentiating terms involving both x and y, the derivative of the y term is dy/dx. As an example, it shows the differentiation of xy using the product rule, which yields y + x*dy/dx. The document concludes by applying this technique to differentiate the equation y4 + xy = x3 - x + 2 implicitly with respect to x.
Composition and inverse of functions
The document discusses composition of functions and inverse functions. It defines composition of functions as combining two functions where one function is performed first and the result is substituted into the second function. The composition is not always commutative. It then provides examples of finding the composition of two functions. Inverse functions are defined as functions where the independent and dependent variables are swapped, and their composition is equal to x. The document demonstrates finding the inverse of functions by swapping variables and checking the composition. It emphasizes that the inverse of a function may not always be a function itself.
CHAIN RULE AND IMPLICIT FUNCTION
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
20 the chain rule
The document discusses general partial derivatives and the chain rule. It defines partial derivatives for functions of multiple variables as holding all other variables constant. It provides an example function z = f(v,w,x,y) and explains how to compute the partial derivatives dz/dx, dz/dy by treating the other variables as constants. The chain rule is introduced for functions of composite variables, where the derivative dz/dt is the sum of the products of partial derivatives along all paths from z to the variable t. An example using this chain rule is worked out in detail. Variable trees are presented as a way to visualize the relationships between composite variables in a chain.
_lecture_05 F_chain_rule.pdf
1. The document discusses the chain rule for functions of several variables. It provides examples of how to use a "tree diagram" to represent variable dependencies and derive the appropriate chain rule statement.
2. It also gives examples of applying the chain rule to find derivatives like df/dt for functions where variables like x, y, and z depend on t, or to find partial derivatives like ∂f/∂s and ∂f/∂t.
3. One example works through applying the chain rule to find df/dt for the function f(x,y) = x^2 + y^2, where x = t^2 and y = t^4, and verifies the
3.7 Inverse Functions
This document discusses inverse functions. It defines an inverse function as undoing another one-to-one function, where applying one function followed by the other results in the original input. The domain of the original function is the range of the inverse, and vice versa. To find the inverse function, interchange the variables of the original function and solve for the new dependent variable. Graphing the inverse results in a reflection of the original function across the line y=x, allowing it to check if the inverse was found correctly. Examples are provided to demonstrate finding and graphing inverse functions.
Partial Differentiation
The document presents information on partial differentiation including:
- Partial differentiation involves a function with more than one independent variable and partial derivatives.
- Notation for partial derivatives is presented.
- Methods for computing first and higher order partial derivatives are explained with examples.
- The concepts of homogeneous functions and the chain rule for partial differentiation are defined.
Calculus
This document discusses the difference between ordinary derivatives and partial derivatives. Ordinary derivatives are taken with respect to one independent variable, while partial derivatives treat all other variables as constant except for the variable being differentiated. The document provides examples of calculating ordinary and partial derivatives for functions of two variables. It also notes that partial derivatives are used in the chain rule to compute ordinary derivatives for functions with multiple variables.
Limits and derivatives
The document discusses functions and their derivatives. It defines functions, different types of functions, and notation used for functions. It then covers the concept of limits, theorems on limits, and limits at infinity. The document defines the slope of a tangent line to a curve and increments. It provides definitions and rules for derivatives, including differentiation from first principles and various differentiation rules. It includes examples of finding derivatives using these rules and taking multiple derivatives.
Understanding the "Chain Rule" for Derivatives by Deriving Your Own Version
Because the Chain Rule can confuse students as much as it helps them solve real problems, we put ourselves in the shoes of the mathematicians who derived it, so that students may understand the motivation for the rule; its limitations; and why textbooks present it in its customary form. We begin by finding the derivative of sin2x without using the Chain Rule. That exercise, having shown that even a comparatively simple compound function can be bothersome to differentiate using the definition of the derivative as a limit, provides the motivation for developing our own formula for the derivative of the general compound function g[f(x)]. In the course of that development, we see why the function f must be continuous at any value of x to which the formula is applied. We finish by comparing our formula to that which is commonly given.
5.2 the substitution methods
The document discusses the substitution method of integration. It explains that while the derivative of an elementary function is another elementary function, the antiderivative may not be. There are two main integration methods: substitution and integration by parts. Substitution reverses the chain rule by letting u be a function of x with derivative u', then substituting u for x and replacing dx with du/u' in the integral.
Relations and functions
The document defines relations and functions. A relation is a set of ordered pairs where each element in the domain (set of x-values) is paired with an element in the range (set of y-values). A function is a special type of relation where each element of the domain is mapped to exactly one element in the range. The document provides examples of relations that are and are not functions based on this one-to-one mapping property. It also discusses using function notation and evaluating functions for different inputs. Finally, it explains how to determine the domain of a function by identifying values that would result in illegal operations like division by zero.
02 first order differential equations
This document discusses various methods for solving first order differential equations, including:
1. Variable separable methods where the equation can be written as a function of x multiplied by a function of y.
2. Homogeneous equations where both sides are homogeneous functions of the same degree.
3. Exact equations where there exists an integrating factor.
4. Equations that can be transformed to an exact or separable form through substitution.
5. Linear equations that can be solved using an integrating factor that is a function of x.
The Chain Rule Powerpoint Lesson
a powerpoint calculus lesson on "The Chain Rule". This is the initial step of an ongoing lesson that we will add to as the class progresses.
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
The document discusses the chain rule for derivatives. It begins by defining function composition and provides examples of composing linear functions. It then states the chain rule theorem, which says that the derivative of a composition is the product of the individual function derivatives evaluated at the same point. Several examples are worked out applying the chain rule to find the derivative of various compositions of functions.
4.1 Inverse Functions
Review composition of functions and one-to-one functions
Identify inverse functions
Write the equation of an inverse function
Ch07
This document discusses inverse functions, including exponential, logarithmic, and inverse trigonometric functions. It begins by defining an inverse function as two functions f and g where g(f(x)) = x and f(g(y)) = y. It then discusses how to find the inverse of a function by solving an equation like y = f(x) for x in terms of y. For a function to have an inverse, it must assign distinct outputs to distinct inputs. The document provides examples of finding inverses and discusses domains, ranges, and interpretations of inverse functions.
Calc 4.5a
This document discusses techniques for evaluating indefinite integrals using substitution, including pattern recognition, change of variables, and the general power rule. It explains that substitution helps evaluate more complex integrals that involve a composite function. Several examples demonstrate identifying the pattern f(g(x))g'(x) and performing u-substitutions to evaluate the integral. It also discusses the need to sometimes multiply and divide by a constant or perform a formal change of variables if the pattern does not exactly match.
Calc 4.5a
This document discusses techniques for integration using substitution. It explains that substitution allows for the integration of more complex composite functions by breaking them into an inside and outside part. The main techniques are pattern recognition, where the integrand fits the standard pattern of f(g(x))g'(x), and change of variables, where a substitution u is made to simplify the integrand. It provides examples of applying both pattern recognition and change of variables substitutions to evaluate definite and indefinite integrals.
Applications of partial differentiation
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
Similar to Chain rule (20)
Understanding the "Chain Rule" for Derivatives by Deriving Your Own Version
Understanding the "Chain Rule" for Derivatives by Deriving Your Own Version
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
lesson10-thechainrule034slides-091006133832-phpapp01.pptx
Recently uploaded
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...Nguyen Thanh Tu Collection
https://app.box.com/s/tacvl9ekroe9hqupdnjruiypvm9rdaneTraditional Musical Instruments of Arunachal Pradesh and Uttar Pradesh - RAYH...
Traditional Musical Instruments of Arunachal Pradesh and Uttar Pradesh
Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptx
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
ANATOMY AND BIOMECHANICS OF HIP JOINT.pdf
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Temple of Asclepius in Thrace. Excavation results
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Leveraging Generative AI to Drive Nonprofit Innovation
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Gender and Mental Health - Counselling and Family Therapy Applications and In...
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
How to Create a More Engaging and Human Online Learning Experience
How to Create a More Engaging and Human Online Learning Experience Wahiba Chair Training & Consulting
Wahiba Chair's Talk at the 2024 Learning Ideas Conference. Wound healing PPT
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Make a Field Mandatory in Odoo 17
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Recently uploaded (20)
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
BÀI TẬP BỔ TRỢ TIẾNG ANH LỚP 9 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2024-2025 - ...
Traditional Musical Instruments of Arunachal Pradesh and Uttar Pradesh - RAYH...
Traditional Musical Instruments of Arunachal Pradesh and Uttar Pradesh - RAYH...
B. Ed Syllabus for babasaheb ambedkar education university.pdf
B. Ed Syllabus for babasaheb ambedkar education university.pdf
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptx
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptx
Leveraging Generative AI to Drive Nonprofit Innovation
Leveraging Generative AI to Drive Nonprofit Innovation
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...
How to Create a More Engaging and Human Online Learning Experience
How to Create a More Engaging and Human Online Learning Experience
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP
Chain rule
- 2. The chain rule basically deals with functions within a function. There is a strong relationship between them. If you consider the links on a chain, one link overlaps the other. If there is a change to either the inside or outside link it affect the relationship between them.
- 3. I believe so far that you have covered derivatives of functions. For example y =f(x) and dy/dx = f’(x). But now considertwo functions f(x) and g(x). We will loo at a function within a function. For instance y = f(g(x)). So then what is y’? The chain rule is what is applied!!
- 4. If y = f(g(x)) y’= f ’(g(x)) . g’(x) OR Alternatively You can let u = g(x) and so it becomes y(u) And then to determine y’, we get y’ = dy/dx = dy/du * du/dx.
- 5. It can also be written as y’ = dy/dx = du/dx * dy/du
- 6. A simple example could be y = (2x +5)2 Let u = 2x +5 therefore y =u2 du/dx = 2 + 0 =2 dy/du = 2u1 = 2(2x +5) Therefore dy/dx = dy/du *du/dx = 2*2(2x +5)=4(2x + 5) = 8x + 20
- 7. y = 5/(3x-5)3 y’ = 5(3x-5)-3 y’ =-3*5(3x-5)-4 *(3) y’ =-45(3x-5)-4 y’ =-45/(3x-5)4
- 8. y = sin(x7) y’ = 7x6*cos(x7)