Question

How do you use the definition of a derivative to find the derivative of `f(x) = (4+x) / (1-4x)`?

Answer 1

`f'(x)=17/(1-4x)^2`

Explanation:

The definition of the derivative states that the derivative of the function `f(x)` equals

`f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h`

Thus, when `f(x)=(4+x)/(1-4x)`, we see that

`f'(x)=lim_(hrarr0)((4+x+h)/(1-4x-4h)-(4+x)/(1-4x))/h`

Clear the denominators from the fraction.

`f'(x)=lim_(hrarr0)(((4+x+h)/(1-4x-4h)-(4+x)/(1-4x))/h)((1-4x-4h)(1-4x))/((1-4x-4h)(1-4x))`

`f'(x)=lim_(hrarr0)((4+x+h)(1-4x)-(4+x)(1-4x-4h))/(h(1-4x-4h)(1-4x))`

Distribute:

`f'(x)=lim_(hrarr0)((-4x^2-4hx-15x+h+4)-(-4x^2-4hx-15x-16h+4))/(h(1-4x-4h)(1-4x))`

Cancel like terms (there are a lot):

`f'(x)=lim_(hrarr0)(17h)/(h(1-4x-4h)(1-4x))`

Cancel the `h` terms:

`f'(x)=lim_(hrarr0)17/((1-4x-4h)(1-4x))`

Plug in `0` for `h`, as the limit can now be evaluated.

`f'(x)=17/((1-4x-0)(1-4x))`

`f'(x)=17/(1-4x)^2`