Question

Calcule f'(x), pela definição: apenas letra f), i), k), j), l)

Answer 1

             Derivada no ponto

$\boxed{ f'(a)=\lim_{x \to \ a}\frac{f(x)-f(a)}{x-a}}$

f)

$f'(1)=\lim_{x \to \ 1}\frac{\frac{1}{{x}^{2}}-1}{x-1}$

$f'(1)=\lim_{x \to \ 1}\frac{\frac{1-{x}^{2}}{{x}^{2}}}{x-1}$

$f'(1)=\lim_{x \to \ 1}(\frac{1-{x}^{2}}{{x}^{2}}).(\frac{1}{x-1})$

$f'(1)=\lim_{x \to \ 1}-(\frac{(x-1)(x+1)}{{x}^{2}}).(\frac{1}{x-1})$

$f'(1)=\lim_{x \to \ 1}-\frac{x+1}{{x}^{2}}$

$\boxed{f'(1)=-2}$

Derivada de uma função pela definição de limite

$\boxed{f'(x)=\lim_{h \to \ 0}\frac{f(x+h)-f(x)}{h}}$

i)

$f'(x)=\lim_{h\to \ 0}\frac{\frac{x+h}{x+h+1}-\frac{x}{x+1}}{h}$

$f'(x)=\lim_{h \to \ 0}\frac{\frac{(x+h)(x+1)-x(x+h+1)}{(x+1)(x+h+1)}}{h}$

$f'(x)=\lim_{h \to \ 0}\frac{\frac{{x}^{2}+x+hx+h-{x}^{2}-hx-x}{(x+1)(x+h+1)}}{h}$

$f'(x)=\lim_{h \to \ 0}\frac{\frac{h}{(x+1)(x+h+1)}}{h}$

$f'(x)=\lim_{h \to \ 0}\frac{1}{h}.\frac{h}{(x+1)(x+h+1)}$

$f'(x)=\lim_{h \to \ 0}\frac{1}{(x+1)(x+h+1)}$

$f'(x)=\frac{1}{(x+1)(x+0+1)}$

$\boxed{f'(x)=\frac{1}{{(x+1)}^{2}}}$

as demais questões são análogas.