Differentiation

Definition of Differentiation
Rules of Differentiation of Functions & Examples Derivatives of Functions
(Simple, Exponential, Logarithmic, Trigonometric and Hyperbolic Functions)
Implicit Differentiation Powerpoint Presentation

Definition of Differentiation

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x.

In other words,

The process of finding a derivative is called Differentiation.

The graph of a function, drawn in blue, and a tangent line to that function, drawn in green. The slope of the tangent line is equal to the derivative of the function at the marked point.

Top


Rules of Differentiation of Functions

Derivative of a constant function

The derivative of f(x) = c where c is a constant is given by

f '(x) = 0


Example:  f(x) = - 10 , then f '(x) = 0



Derivative of a power function (Power rule)

The derivative of f(x) = x r where r is a constant real number is given by

f '(x) = r *x(r-1)


Example:  f(x) = x -2 , then f '(x) = -2 x -3 = -2 / x 3




Derivative of a function multiplied by a constant


The derivative of f(x) = c g(x) is given by

f '(x) = c g '(x)

Example: f(x) = 3x 3 ,
let c = 3 and g(x) = x 3, then f '(x) = c g '(x) = 3 (3x 2) = 9 x 2




Derivative of the sum of functions (Sum rule)


The derivative of f(x) = g(x) + h(x) is given by

f '(x) = g '(x) + h '(x)

Example: f(x) = x2 + 4
let g(x) = x 2 and h(x) = 4, then f '(x) = g '(x) + h '(x) = 2x + 0 = 2x




Derivative of the difference of functions

The derivative of f(x) = g(x) - h(x) is given by

f '(x) = g '(x) - h '(x)

Example: f(x) = x 3 - x -2
let g(x) = x 3 and h(x) = x -2, then
f '(x) = g '(x) - h '(x) = 3 x2 - (-2 x-3) = 3 x 2 + 2x-3




Derivative of the product of two functions(Product rule)

The derivative of f(x) = g(x) h(x) is given by

f '(x) = g(x) h '(x) + h(x) g '(x)


Example:
f(x) = (x 2 - 2x) (x - 2)
let g(x) = (x 2 - 2x) and h(x) = (x - 2), then
f '(x) = g(x) h '(x) + h(x) g '(x) = (x 2 - 2x) (1) + (x - 2) (2x - 2) = x2 - 2x + 2 x 2 - 6x + 4
=3 x2 - 8x + 4




Derivative of the quotient of two functions (Quotient rule)

The derivative of f(x) = g(x) / h(x) is given by

f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x)2


Example: f(x) = (x - 2) / (x + 1)
let g(x) = (x - 2) and h(x) = (x + 1), then
f '(x) = ( h(x) g '(x) - g(x) h '(x) ) / h(x)2
= ( (x + 1)(1) - (x - 2)(1) ) / (x + 1)2 = 3 / (x + 1) 2






Top

Derivatives of Functions


Derivatives of Simple Functions





















Derivatives of Exponential and Logarithmic functions

































Derivatives of Trigonometric functions



























Derivatives of Hyperbolic functions



























Top

Implicit Differentiation

An equation of the form f(x, y) in which y is not expressible directly in terms of x, is known as an implicit function of x and y.

To implicitly differentiate an implicit function follow the steps below.

1. Apply all differentiation rules. Except to y.

Given equation is:

y3 - x2 = -5
y3 - 2x = 0


2. Then differentiate the y's in regard to x. In turn you add a y prime to the differentiated y's

3y2y' -2x = 0

Here, y' is refered to d(y)/d(x)

3. Now factor out any y primes

Hint: In this example there are none.

4. Lastly, place all the y primes on one side

Solution: y' = (2x) / (3y2)





Top
Clicky Web Analytics