Derivatives And It’s Simple Applications

Introduction to derivative

Modern development calculus

The modern development of calculus is usually credited to Isaac Newton and Gottfried Leibniz who provided

independent and unified approaches to differentiation and derivatives

Introduction to derivative

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications.

The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation

Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point.

Derivative of usual function

DERIVATIVES OF USUAL FUNCTIONS

1) Differentiating Constant FunctionsRemember that a constant function has the same value at every point. The graph of such a function is a horizontal line:Now at any point, the tangent line to the graph (remember this is the line which best approximates the graph) is the same horizontal line. Since the derivative measures the slope of the tangent line and a horizontal line has slope zero, we expect the following:Derivative of a constant :

DERIVATIVES OF USUAL FUNCTIONS

2)IDENTITY FUNCTIONLet f(x)=x, the identity function of x then, F’(X)=(X)’=1

3)FUNCTION OF FORM X^NLet f(x)=x^n,a function of x,and n a real constant then, F’(X)=(X^N)’=NX^N-1

4)EXPONENTIAL FUNCTIONIn mathematics, an exponential function is a function of the formF(X)=A^X where a>0 then f’(x)=(a^x)’=a^x(log a)

Basic derivation rules

Basic Derivation RULES1)Constant Multiple RuleThe derivative of a constant multiplied by a function is the constant multiplied by the derivative of the original function:

2)Sum/Difference RulesThe derivative of the sum of two functions is the sum of the derivatives of the two functions

Basic Derivation RULES3)Product RuleThe derivative of the product of two functions is NOT the product of the functions' derivatives; rather, it is described by the equation below:

4)Quotient RuleThe derivative of the quotient of two functions is NOT the quotient of the functions' derivatives; rather, it is described by the equation below:

Let us see some examples to understand how Basic rules of

derivatives are applied

examples of Basic Derivation

Example 1) = 4()=4(2)=8

Example 2) (+ =Example 3)

Example 4) = = = =

Derivative of composite function

DERIVATIVE OF COMPOSITE FUNCTION THE CHAIN RULE

The chain rule is a formula for computing the derivative of the composition of two or more function. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f o gThe function which maps x to f(g(x)) in terms of the derivatives of f and g and the product of function as follows:(f o g)’=(f’ o g).g’

DERIVATIVE OF COMPOSITE FUNCTION

A function inverse can be thought of as the reversal of whatever our function does to its input. Composition of two inverse function results in identity function. Inverse of inverse of a function is that function itselfwhere (f o )(x) = (o f)(x)

Steps to solve inverse function is summarized below.•On both sides of the equation replace x with (x).•Substitute f((x)) = x•Solve (x) in terms of x.

Let us see some examples to understand How composite

function are applied

DERIVATIVE OF COMPOSITE FUNCTION

Example 1: Find the derivative of Sin 10xSolution:Let y = sin u and u = 10x= cos u and = 10

Hence = cos u * 10 = 10 cos 10x

DERIVATIVE OF COMPOSITE FUNCTION

Example 2:The two functions f and g are defined on the set of real numbers such that: f(x)= + 5 and g(x) = x√x. Find fog and gof and show that fog≠gof.Solution :(fog)(x) = f{g(x)} (fog)(x) = f(x√x) = + 5 = x + 5

(gof)(x) = g(f(x) (gof)(x) = g(+ 5) = Therefore fog ≠ gof

Second derivative

Second DERIVATIVE

In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changingfor example, the second derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing with respect to time.In Leibniz notation: where the last term is the second derivative expression.

Second DERIVATIVE

Second derivative testThe relation between the second derivative and the Graph can be used to test whether a stationary point for a Function is a local maximum or a local minimum. SpecificallyIf f’’(x) <0 then f has a local maximum at x.If f’’(x)>0 then f has a local minimum at x.If f’’(x)=0 , the second derivative test says nothing about the point x , a possible inflection point.

Second DERIVATIVE

Second derivative testThe reason the second derivative produces these results can be seen by way of a real-world analogy. Example,Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration.Clearly the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.

Let us see some examples to understand how application of

derivatives takes place

Applications of DERIVATIVEs In physics A particle is moving in such a way that its displacement ‘s’ at a time ‘t’ is given by, find the velocity and acceleration after 2 sec.Solution :Therefore velocity = = 4(2) + 5 =13

Acceleration = a =The velocity and acceleration are 13units per sec and 4 units per sec square.

Applications of DERIVATIVEs A ladder of length 20 feet rests against a smooth vertical wall. The lower end,which is on a smooth horizontal surface is moved away from the wall at rate of 4feet/sec. Find the rate at which the upper end moves when the lower end is 12 feet away from the wall. Solution: Let AB be ladder ,Let OB=x & OA =Y A 4ft/sec From the figure y 2x +2y = 0 O B …………….(1) Now x=12ft. =- y=16 From (1) we get Therefore the upper end is moving downwards at the rate of 3ft/sec.