# Summary

• Basic laws of differentiation

Let f (x) and g(x) be any two functions of x. Then,
• Scalar multiple rule

where c is any constant
• Sum and difference rule

• Product rule

• Quotient rule

• Derivative of a function of a function

Let y = f (u), where u = g(x)

Then
• Implicit functions

A function f (xy) = 0 is said to be an implicit function, if y cannot be directly defined as a function of x.

In such a case, we will differentiate both sides of this equation w.r.t x, collect the terms containing  on one side; transfer other terms to the other side and divide by the coefficient of   to get its value.
• Parametric equation

If x = f (t) and y = g(t), then
• Logarithmic differentiation

When a function is expressed in any of the following forms, its derivative can be obtained by taking the logarithm of the function and then differentiating it.

(i) A product of a number of functions

(ii) When a function is raised to some exponent which is also a function

(iii) A number of functions are divided

This method of differentiation is known as logarithmic differentiation.
• Higher order derivatives

Let y = f (x), be a function of x is called the first derivative of y with respect to x.

The derivative of f â€²(x) is called the second derivative of y with respect to x,

i.e.
• Integral Calculus

Integration is the inverse operation of differentiation.

If the derivative of y with respect to x is given by , the integral of f â€²(x) with respect to x is given by
• Basic laws of integration

Let f (x) and g(x) be any two functions of x, then

# Integration by Substitution

Sometimes it is not possible to be able to integrate f (x) directly. We can substitute f (x) into some other function g (t) to make it readily integrable.

# Integration by Parts

Let u = f (x) and v = g(x) be two different functions of x. Then

The function that can be easily integrated should be chosen as v and the other function which is easily differentiable should be chosen as u.

# Integration by the Method of Partial Fraction

Type 1:

where A and B are constants to be determined.

Type 2:

where AB and C are constants to be determined.

Type 3:

where A and B and C are constants to be determined.

# Definite Integration

Let  where f (x) is the integral of F(x).

As x changes from a to b, the value of the integral changes from f (a) to f (b). This can be shown as

# Properties of Definite Integrals

•
•
•
•
•
•  is an even function
•  if f (x) is an odd function