The Chain Rule Powerpoint Lesson
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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.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
Lesson 4 Nov 17 09
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- Rules for finding derivatives of sums, products, and quotients using the product, quotient, and chain rules
- Derivatives of common trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant
- An exercise involving 23 problems calculating derivatives
Lesson 12: The Product and Quotient Rule
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Lesson 9: The Product and Quotient Rule
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125 7.7
This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
Recommended
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.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f multiplied by the derivative of the inner function g. This can be written as f'(g(x)) * g'(x). Several examples are provided to demonstrate how to use the chain rule to take the derivative of various composite functions.
Lesson 4 Nov 17 09
The document provides information on derivatives including:
- Rules for finding derivatives of sums, products, and quotients using the product, quotient, and chain rules
- Derivatives of common trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant
- An exercise involving 23 problems calculating derivatives
Lesson 12: The Product and Quotient Rule
The product rule shows us how to take the derivative of a product of functions. Ditto the quotient rule.
Lesson 9: The Product and Quotient Rule
These rules allow us to differentiate the product or quotient of functions whose derivatives are themselves known.
Class 21: Changing State
This document discusses changing state in programs through mutation. It introduces mutable data structures like mutable cons cells that can be modified using procedures like set-mcar! and set-mcdr!. Mutation allows the value associated with a name to change, violating the substitution model of evaluation. Evaluation order matters since values may change. Programmers must be careful to consider when things happen with mutation, as order affects results. The homework assignment is to read through the end of Chapter 9 in preparation for revising the evaluation rules on Friday to handle state changes through mutation.
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This document discusses the chain rule for finding derivatives. It explains that the chain rule is needed when taking the derivative of a composition of functions, where an "inside function" is plugged into an "outside function". The chain rule formula is given as the derivative of the outside function multiplied by the derivative of the inside function. Several examples are worked through, applying the chain rule when the power rule alone cannot be used, such as when the base of an exponent is a function rather than a variable. The document also notes that problems may require using multiple derivative rules, like the product rule and chain rule, to fully solve them.
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- It is more systematic and avoids mistakes from expanding products
- It works for any differentiable functions u and v, not just polynomials
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The chain rule is used to find the derivative of composite functions, where an inner function is composed inside an outer function. It states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x), where f'(g(x)) is the derivative of the outer function with respect to the inner function, and g'(x) is the derivative of the inner function. Examples show using the chain rule to take derivatives of functions like y = (3x+1)^7, where the inner function is 3x+1 and the outer function is the exponent. The document also provides examples of when the chain rule is and isn't needed.
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The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outside function f'(g(x)) multiplied by the derivative of the inside function g'(x). This allows the calculation of derivatives of more complex functions that cannot be solved using basic derivative rules. Several examples are provided to demonstrate how to use the chain rule to calculate derivatives of various composite functions.
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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.
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The document defines key concepts related to derivatives including:
1) Notations for derivatives such as F'(x), dy/dx, Dx f(x), and Ϋ.
2) Rules for derivatives including the power rule, product rule, and quotient rule.
3) Examples of applying the rules to find the derivatives of various functions such as f(x) = 4x, f(x) = (x3 - 2x)(x5 + 6x2), and f(x) = (x+2)/x3.
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The document outlines various derivative rules including:
- The power rule for derivatives of functions with exponents like x^n.
- Sum and difference rules for derivatives of sums and differences of functions.
- Product and quotient rules for derivatives of products and quotients of functions.
- The chain rule for derivatives of composite functions like f(g(x)).
- Derivative rules for common trigonometric, exponential and logarithmic functions.
Examples are provided to demonstrate how to apply each rule to find derivatives.
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Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
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The document discusses the chain rule and how to use it to differentiate and integrate composite functions. The chain rule states that if h(x) = g(f(x)), then h'(x) = g'(f(x))f'(x). It provides examples of applying the chain rule to differentiate functions like sin(x2 - 4) and integrate functions like ∫(3x2 + 4)3 dx. It also discusses how to integrate functions of the form f'(x)g(f(x)) by recognizing them as derivatives of composite functions.
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3) Additional examples are provided of using the chain rule to find derivatives of more complex expressions involving radicals, quotients, and other functions.
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- Notation for partial derivatives
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So in this example, using the product rule is preferable to direct multiplication.
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The Chain Rule Powerpoint Lesson
The document explains the chain rule, which provides a method for finding the derivative of a composite function. The chain rule states that the derivative of a composite function f(g(x)) is equal to the derivative of the outside function f'(g(x)) multiplied by the derivative of the inside function g'(x). This allows the calculation of derivatives of more complex functions that cannot be solved using basic derivative rules. Several examples are provided to demonstrate how to use the chain rule to calculate derivatives of various composite functions.
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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.
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The Lagrange multiplier method provides a strategy for finding the maxima and minima of a function subject to constraints. It involves setting up a system of equations involving the function, its derivatives, and the constraints and their derivatives. Solving this system of equations yields candidate maxima/minima points, which are then checked in the original function to determine if they are actually maxima or minima. The document provides examples of applying the Lagrange multiplier method to problems with single and multiple constraints.
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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.
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The document defines key concepts related to derivatives including:
1) Notations for derivatives such as F'(x), dy/dx, Dx f(x), and Ϋ.
2) Rules for derivatives including the power rule, product rule, and quotient rule.
3) Examples of applying the rules to find the derivatives of various functions such as f(x) = 4x, f(x) = (x3 - 2x)(x5 + 6x2), and f(x) = (x+2)/x3.
Derivatives vinnie
The document outlines various derivative rules including:
- The power rule for derivatives of functions with exponents like x^n.
- Sum and difference rules for derivatives of sums and differences of functions.
- Product and quotient rules for derivatives of products and quotients of functions.
- The chain rule for derivatives of composite functions like f(g(x)).
- Derivative rules for common trigonometric, exponential and logarithmic functions.
Examples are provided to demonstrate how to apply each rule to find derivatives.
01. Differentiation-Theory & solved example Module-3.pdf
Total No. of questions in Differentiation are-
In Chapter Examples 31
Solved Examples 32
The rate of change of one quantity with respect to some another quantity has a great importance. For example the rate of change of displacement of a particle with respect to time is called its velocity and the rate of change of velocity is
called its acceleration.
The following results can easily be established using the above definition of the derivative–
d
(i) dx (constant) = 0
The rate of change of a quantity 'y' with respect to another quantity 'x' is called the derivative or differential coefficient of y with respect to x.
Let y = f(x) be a continuous function of a variable quantity x, where x is independent and y is
(ii)
(iii)
(iv)
(v)
d
dx (ax) = a
d (xn) = nxn–1
dx
d ex =ex
dx
d (ax) = ax log a
dependent variable quantity. Let x be an arbitrary small change in the value of x and y be the
dx
d
(vi) dx
e
(logex) = 1/x
corresponding change in y then lim
y
if it exists, d 1
x0 x
is called the derivative or differential coefficient of y with respect to x and it is denoted by
(vii) dx
(logax) =
x log a
dy . y', y
dx 1
or Dy.
d
(viii) dx (sin x) = cos x
So, dy dx
dy
dx
lim
x0
lim
x0
y
x
f (x x) f (x)
x
(ix) (ix)
(x) (x)
d
dx (cos x) = – sin x
d (tan x) = sec2x
dx
The process of finding derivative of a function is called differentiation.
If we again differentiate (dy/dx) with respect to x
(xi)
d (cot x) = – cosec2x
dx
d
then the new derivative so obtained is called second derivative of y with respect to x and it is
Fd2 y
(xii) dx
d
(xiii) dx
(secx)= secx tan x
(cosec x) = – cosec x cot x
denoted by
HGdx2 Jor y" or y2 or D2y. Similarly,
d 1
we can find successive derivatives of y which
(xiv) dx
(sin–1 x) = , –1< x < 1
1 x2
may be denoted by
d –1 1
d3 y d4 y
dn y
(xv) dx (cos x) = –
,–1 < x < 1
dx3 ,
dx4 , ........, dxn , ......
d
(xvi) dx
(tan–1 x) = 1
1 x2
Note : (i)
y is a ratio of two quantities y and
x
(xvii) (xvii)
d (cot–1 x) = – 1
where as dy
dx
dy
is not a ratio, it is a single
dx
d
(xviii) (xviii)
(sec–1 x) =
1 x2
1
|x| > 1
quantity i.e.
dx dy÷ dx
dx x x2 1
(ii)
dy is
dx
d (y) in which d/dx is simply a symbol
dx
(xix)
d (cosec–1 x) = – 1
dx
of operation and not 'd' divided by dx.
d
(xx) dx
(sinh x) = cosh x
d
(xxi) dx
d
(cosh x) = sinh x
Theorem V Derivative of the function of the function. If 'y' is a function of 't' and t' is a function of 'x' then
(xxii) dx
d
(tanh x) = sech2 x
dy =
dx
dy . dt
dt dx
(xxiii) dx
d
(xxiv) dx
d
(coth x) = – cosec h2 x (sech x) = – sech x tanh x
Theorem VI Derivative of parametric equations If x = (t) , y = (t) then
dy dy / dt
=
(xxv) dx
(cosech x) = – cosec hx coth x
dx dx / dt
(xxvi) (xxvi)
(xxvii) (xxvii)
d (sin h–1 x) =
The chain rule
The document discusses the chain rule and how to use it to differentiate and integrate composite functions. The chain rule states that if h(x) = g(f(x)), then h'(x) = g'(f(x))f'(x). It provides examples of applying the chain rule to differentiate functions like sin(x2 - 4) and integrate functions like ∫(3x2 + 4)3 dx. It also discusses how to integrate functions of the form f'(x)g(f(x)) by recognizing them as derivatives of composite functions.
Benginning Calculus Lecture notes 5 - chain rule
1) The document discusses the chain rule and higher derivatives in calculus. It defines the chain rule and provides examples of applying it to find the derivative of composite functions.
2) It also explains how to take higher derivatives by applying the derivative operator multiple times, and gives an example of finding the nth derivative of xn.
3) Additional examples are provided of using the chain rule to find derivatives of more complex expressions involving radicals, quotients, and other functions.
Jan. 4 Function L1
This document defines and provides examples of functions. It discusses:
- Functions are relations where each input has exactly one output
- The vertical line test to determine if a relation is a function
- Common operations on functions like addition, subtraction, multiplication, and division
- Composite functions which take the output of one function as the input of another
- Examples of evaluating composite functions and performing operations on functions
Partial diferential good
This document provides an introduction to partial differentiation, including:
- Defining partial derivatives and how they are calculated by treating all but one variable as a constant
- Examples of finding partial derivatives using the product, quotient, and chain rules
- Higher order partial derivatives and mixed partial derivatives
- Notation for partial derivatives
- A quiz on partial derivatives concepts
Day_1_-_Rules_for_Differentiation (1).ppt
The document discusses rules for differentiation, including the power rule, constant rule, constant multiple rule, and sum and difference rules. It explains that differentiation is the process of computing the derivative of a function, which represents the instantaneous rate of change and slope of the tangent line. Examples are provided to demonstrate applying each rule to find the derivative of various functions.
Hyperfunction method for numerical integration and Fredholm integral equation...
The slide of a speech in the conference "Computational Methods and Function Theory 2017" held in Lublin, Poland in July 2017.
Rules_for_Differentiation.ppt
The document discusses rules for differentiation, including the power rule, constant rule, constant multiple rule, and sum and difference rules. It explains that the power rule states that the derivative of x^N is Nx^(N-1), the constant rule is that the derivative of a constant is 0, and the constant multiple rule is that the derivative of c*f(x) is c*f'(x). It also explains that the sum and difference rules state that the derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives. Several examples are worked out applying these rules to find derivatives of various functions.
6.2 the indefinite integral
This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
6.2 the indefinite integral
This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
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.
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.
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
ここ3000字までしか入らないけどタイトルの方がたくさん文字入ると思います。
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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The Chain Rule Powerpoint Lesson
- 2. The Problem Complex Functions Why? not all derivatives can be found through the use of the power, product, andquotient rules
- 3. Working To A Solution Composite function f(g(x)) f is the outside function, g is the inside function z=g(x), y=f(z), and y=f(g(x)) Therefore, a small change in x leads to a small change in z a small change in z leads to a small change in y
- 4. Working To A Solution cont. Therefore, (∆y/∆x) = (∆y/∆z) (∆z/∆x) Since (dy/dx)=limx0 (∆y/∆x) (dy/dx) = (dy/dz) (dz/dx) This is known as “The Chain Rule”
- 6. Example:[ (x^3) + 2x + 1 ]^3 inside function = z = g(x) = (x^3) + 2x + 1 outside function = f(z) = z^3 g’(x) = (3x^2) + 2 f’(g(x)) = 3 [ (x^3) + 2x + 1 ]^2 (d/dx) f(g(x)) = 3 [ (3x^2) + 2 ] [ (x^3) + 2x + 1 ]^2 = [ (9x^2) + 6 ] [ (x^3) + 2x + 1 ]^2
- 7. Examplee^(4x^2) inside function = z = g(x) = 4x^2 outside function = f(z) = e^z g’(x) = 8x f’(z) = e^z (d/dx) f(g(x)) = 8xe^z = 8xe^(4x^2)
- 8. Sources Slideshow created using Microsoft PowerPoint Clipart and themes supplied through Microsoft PowerPoint Mathematics reference and notation from “Calculus: Single and Multivariable” 4th edition, by Hughes-Hallet|Gleason|McCallum|et al.