The following rules are useful in finding derivatives using formula for the various functions:

**Rule 1**: If f(x) = x^{n}, then f'(x) = nx^{n-1}.

Or if y = x^{n}, then y’ or or = nx^{n-1.}

**Rule 2**: If f(x) = kx^{n}, then f'(x) = knx^{n-1}.

Or if y = kx^{n}, then y’ or or = nkx^{n-1}.

In general, = k

So = k f'(x)

**Rule 3**: When an equation contains the sum of two functions, then the derivative of this equation is the sum of the derivatives of the two functions.

= +

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

**Proof for this rule**:

=

=

=

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

**Rule 4**: Similarly the derivation of the difference of two equations is the difference of the derivatives of the two functions.

= –

= f'(x) – g'(x)

**Proof for this rule**:

=

=

=

= f'(x) – g'(x)

**Rule 5**: The derivative of the product of two functions – known as **Product Rule** – is given by

= +

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

**Rule 6**: The derivative of the quotient of two functions – known as **Quotient Rule** – is given by

=

This rule is a modification of the Product Rule by writing

as

= + f(x)

= f'(x).(g(x))^{-1} − f(x).(g(x))^{-2}.g'(x)

= –

=

In simplistic terms, we can look at the **Product Rule** and **Quotient Rule** as follows:

Let u = f(x) and v = g(x).

The product rule can be expressed as (uv) = u’v + uv’, and

The quotient rule can be expressed as =

These formulae are referred as **Leibnitz rule** for differentiation of product and quotient of functions respectively.

Now let us look at some examples of differentiation using these formulae.