Question

Use the four-step definition of the derivative to find f'(x) if f(x) = −4x^3 −1.

f(x+h) =

f(x+h)-f(x) =

f(x+h)-f(x)/h =

f'(x) =

Answer 1

`\stackrel{de finition \textit{ of a derivative as a limit}}{\lim\limits_(h\to 0)~\cfrac{f(x+h)-f(x)}{h}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{[-4(x+h)^3-1]~~ - ~~[-4x^3-1]}{h} \\\\\\ \cfrac{[-4(x^3+3x^2h+3xh^2+h^3)-1]~~ ~~+4x^3+1}{h} \\\\\\ \cfrac{[-4x^3-12x^2h-12xh^2-4h^3-1]~~ ~~+4x^3+1}{h} \\\\\\ \cfrac{-4x^3-12x^2h-12xh^2-4h^3-1+4x^3+1}{h}\implies \cfrac{-12x^2h-12xh^2-4h^3}{h}`

`\cfrac{h(-12x^2-12xh-4h^2)}{h}\implies -12x^2-12xh-4h^2 \\\\\\ \lim\limits_(h\to 0)~-12x^2-12xh-4h^2\implies \lim\limits_(h\to 0)~-12x^2-12x(0)-4(0)^2 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \lim\limits_(h\to 0)~-12x^2~\hfill`