Question

Calcule a derivada de 4x2+1 por definição

Answer 1

Resposta:

Dada a função:

$f(x) = 4x^{2} + 1,$

calculemos sua derivada:

$\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\\\\=\lim_{h \to 0} \frac{[4(x+h)^{2}+1]-(4x^{2}+1)}{h}\\\\= \lim_{h \to 0} \frac{4x^{2}+8xh+4h^{2}+1-4x^{2}-1}{h}\\\\= \lim_{h \to 0} \frac{8xh+h^{2}}{h}\\\\= \lim_{h \to 0} \frac{h(8x + h)}{h}\\\\= \lim_{h \to 0} (8x + h)\\\\= 8x + 0\\\\= 8x.$