How do you use the definition of a derivative to find the derivative of f(x) = (4+x) / (1-4x)f(x)=4+x14x?

1 Answer
Apr 23, 2016

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