9. Basic Concepts of Differential and Integral Calculus

DifferentiationDifferentiation is the process of finding rate of change of a dependent variable with respect to independent variable.

Differentiation Formulae

( )y f x

Dependent Variable

Independent Variable

Also remember the following formulae:

Differentiation Techniques

1.

Exp:

2.

Exp:

3.

Exp:

4. Product Rule:

In general,

Exp:

5. Quotient Rule:

Exp:

6. Derivative of a function of function (Chain Rule):

If

Exp: If , then find .

Solution: Let .

7. Derivative of Implicit Functions:

A function in the form of e.g. where y cannot be directly defined as a function of x is called an implicit function of x.

Exp: If , then find .

Solution:

8. Derivative of Parametric Equation:When both the variables are expressed in terms of a parameter (a third variable), then the involved equations are called as parametric equations.

Exp: If , then find .

Solution:

9. Logarithmic Differentiation:This procedure of finding out derivative by taking logarithm is used in the following two situations:

.

When the function is the product of number of functions.

Exp: If , then find .

Solution:

X

Y

O

P(x, y)

Exp: If , then find .

Solution:

Higher Order Differentiation:

Exp: If , then find .

Solution:

Geometric Meaning of the Derivative

The derivative of at a point x represents the slope or gradient of the tangent to the curve at the point x.

Integration

Integration is the reverse process of differentiation.

Integration Formulae

( )f x f x

Differentiation

Integration

Also remember the following formulae:

Integration by parts

Priority of functions to be considered as a first function is as follows:Logarithmic Functions e.g. Algebric Functions e.g. Exponential Functions e.g.

Definite Integration

Important Property of definite Integral