Question

Como faço pra calcular f`(x), pela definição?
lim h->0 f(x+h)-f(x)/h

f(x)=5x

Answer 1

A saber, $f(x)=5x\Leftrightarrow\,f(x+h)=5(x+h)$.

Segue que,

$f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\\\\\\f'(x)=\lim_{h\rightarrow0}\frac{5(x+h)-5x}{h}\\\\\\f'(x)=\lim_{h\rightarrow0}\frac{5x+5h-5x}{h}\\\\\\f'(x)=\lim_{h\rightarrow0}\frac{5h}{h}\\\\\\f'(x)=\lim_{h\rightarrow0}5\\\\\boxed{f'(x)=5}$