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# Finding Derivatives using Formula

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

Rule 1: If f(x) = xn, then f'(x) = nxn-1.

Or if y = xn, then y’  or  or   = nxn-1.

Rule 2: If f(x) = kxn, then f'(x) = knxn-1.

Or if y = kxn, then y’  or  or   = nkxn-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.