Here are some more examples of derivatives of functions, obtained using the first principles of differentiation.

**Example 1**: Find the derivative of f(x) = 6 using first principles.

f(x + h) = 6

f'(x) =

=

= 0 (as )

**Remember**: In general if f(x) = c, where c is a fixed real number, then f'(x) = 0.

**Example 2**: Find the derivative of f(x) = x^{3}

f(x + h) = (x + h)^{3}

= x^{3} + 3x^{2}h + 3xh^{2} + h^{3}

f(x + h) – f(x) = (x^{3} + 3x^{2}h + 3xh^{2} + h^{3}) –^{ }x^{3}

= 3x^{2}h + 3xh^{2} + h^{3}

f'(x) =

=

=

= 3x^{2}

**Remember**: If we have an equation with power in it, the derivative of the equation reduces the power index by 1, and the function’s power becomes the co-efficient of the derivative function.

In other words, if f(x) = x^{n}, then f'(x) = nx^{n-1}

The above generalisation will hold for negative powers also.

**Example 3**: Find the derivative of f(x) =

f(x + h) =

f'(x) =

=

=

=

=

=

So derivative of f(x) =

**Example 4**: Find the derivative of f(x) =

f(x + h) =

f'(x) =

=

=

= using a^{2} – b^{2} = (a-b)(a+b)

=

= –

= –

= –

So when f(x) = , f'(x) =

**Remember**: In general, if f(x) = , f'(x) =

and, if f(x) = x^{-n}, f'(x) = -n(x)^{-n-1} =

**To summarise** –

- If f(x) = c, where c is a fixed real number, then f'(x) = 0.
- If f(x) = x
^{n}, then f'(x) = nx^{n-1} - If f(x) = , f'(x) =
- If f(x) = x
^{-n}, f'(x) = -n(x)^{-n-1}=

We now look at finding derivatives using formula