How do you use the definition of a derivative to find the derivative of f(x) = (4+x) / (1-4x)f(x)=4+x1−4x?
1 Answer
Apr 23, 2016
Explanation:
The definition of the derivative states that the derivative of the function
f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h
Thus, when
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
f'(x)=lim_(hrarr0)17/((1-4x-4h)(1-4x))
Plug in
f'(x)=17/((1-4x-0)(1-4x))
f'(x)=17/(1-4x)^2