Question

Calcule a derivada usando a definição; f(x)= x²-3x para x=2

Answer 1

A derivada da função f(x) pode ser calculada pela definição:

$\underset{h\to0}{lim}~~\frac{f(x+h)-f(x)}{h}$

Substituindo a função dada no limite, temos:

$\underset{h\to0}{lim}~~\frac{\left[(x+h)^2-3.(x+h)\right]~-~\left[x^2-3x\right]}{h}~=\\\\\\=~\underset{h\to0}{lim}~~\frac{\left[x^2+2xh+h^2-3x-3h\right]~-~\left[x^2-3x\right]}{h}$

$=~\underset{h\to0}{lim}~~\frac{x^2+2xh+h^2-3x-3h-x^2+3x}{h}\\\\\\=~\underset{h\to0}{lim}~~\frac{2xh+h^2-3h}{h}\\\\\\=~\underset{h\to0}{lim}~~\frac{h.(2x+h-3)}{h}$

$=~\underset{h\to0}{lim}~~\frac{h\!\!\!\backslash.(2x+h-3)}{h\!\!\!\backslash}\\\\\\=~\underset{h\to0}{lim}~~(2x+h-3)\\\\\\=~2x+0-3\\\\\\=~\boxed{2x-3}$

Por fim, podemos calcular a derivada no ponto x=2:

$f'(2)~=~2\,.\,(2)~-~3\\\\\\f'(2)~=~4-3\\\\\\\boxed{f'(2)~=~1}$