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# More examples of derivatives

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) = x3

f(x + h)  =  (x + h)3

=  x3 + 3x2h + 3xh2 + h3

f(x + h) – f(x)  =  (x3 + 3x2h + 3xh2 + h3) – x3

=  3x2h + 3xh2 + h3

f'(x)  =

=

=

=  3x2

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)  =  xn, then f'(x)  =  nxn-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 a2 – b2 = (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

1. If f(x) = c, where c is a fixed real number, then f'(x) = 0.
2. If f(x)  =  xn, then f'(x)  =  nxn-1
3. If f(x) = , f'(x) =
4. If f(x) = x-n, f'(x)  =  -n(x)-n-1  =

We now look at finding derivatives using formula