This document discusses the topic of differentiation. It begins by defining differentiation and listing some fundamental rules, such as the chain rule and differentiation of constants. It then discusses geometrically what the derivative represents at a point and lists several types of differentiable functions. The document goes on to explain differentiation using substitution, of implicit functions, and of parametric functions. It also covers successive differentiation, Leibnitz's theorem, and differentiation of special function types. The document provides an overview of differentiation concepts and rules.
Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
Application of definite integrals,we will explore some of the many application of definite integral by using it to calculate areas between two curves, volumes, length of curves, and several other application.
Integration Made Easy!
The derivative of a function can be geometrically interpreted as the slope of the curve of the mathematical function f(x) plotted as a function of x. But its implications for the modeling of nature go far deeper than this simple geometric application might imply. After all, you can see yourself drawing finite triangles to discover slope, so why is the derivative so important? Its importance lies in the fact that many physical entities such as velocity, acceleration, force and so on are defined as instantaneous rates of change of some other quantity. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity.
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A presentation done as a part of the course Semester 5 course Environmental Studies during my Under-Graduate course in Engineering.
The presentation describes the impact of Information Technology on Environment and Human Health systematically laying emphasis particularly on the environmental aspects
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
01. Differentiation-Theory & solved example Module-3.pdfRajuSingh806014
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 Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
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The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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7. (v) if d / d(x) f(x) = f(x), then d / d(x) f(ax + b) = a f(ax + b)
(vi) Differentiation of a constant function is zero i.e., d / d(x) (c) = 0.
Geometrically Meaning of Derivative at a Point
Geometrically derivative of a function at a point x = c is the slope of the
tangent to the curve y = f(x) at the point {c, f(c)}. Slope of tangent at P =
lim x → c f(x) – f(c) / x – c = {df(x) / d(x)} x = c or f’ (c).
Different Types of Differentiable Function
1. Differentiation of Composite Function (Chain Rule)
If f and g are differentiable functions in their domain, then fog is also
differentiable and (fog)’ (x) = f’ {g(x)} g’ (x) More easily, if y = f(u) and u
= g(x), then dy / dx = dy / du * du / dx. If y is a function of u, u is a
function of v and v is a function of x. Then, dy / dx = dy / du * du / dv *
dv / dx.
Cont….
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8. 2. Differentiation Using Substitution
3. Differentiation of Implicit Functions
If f(x, y) = 0, differentiate with respect to x and collect the terms containing dy /
dx at one side and find dy / dx. Shortcut for Implicit Functions For Implicit
function, put d /dx {f(x, y)} = – ∂f / ∂x / ∂f / ∂y, where ∂f / ∂x is a partial
differential of given function with respect to x and ∂f / ∂y means Partial
differential of given function with respect to y.
Cont…
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9. 4. Differentiation of Parametric Functions
If x = f(t), y = g(t), where t is parameter, then dy / dx = (dy / dt) / (dx / dt) = d
/ dt g(t) / d / dt f(t) = g’ (t) / f’ (t)
5. Differential Coefficient Using Inverse Trigonometrical Substitutions
Sometimes the given function can be deducted with the help of inverse
Trigonometrical
substitution and then to find the differential coefficient is very easy.
Cont…
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11. Differentiation of a Function with Respect to Another Function
Let y = f(x) and z = g(x), then the differentiation of y with respect to z is
dy / dz = dy / dx / dz / dx = f’ (x) / g’ (x)
Successive Differentiations
If the function y = f(x) be differentiated with respect to x, then the result
dy / dx or f’ (x), so obtained is a function of x (may be a constant).
Cont…
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12. Leibnitz Theorem
If u and v are functions of x such that their nth derivative exist, then
nth Derivative of Some Functions
Cont…
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