Question

Пользуясь определением, найдите производную функции : 1)у= х^2 2)у= х^3

Answer 1

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

2)
$\displaystyle \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}= \lim_{h \to 0} \frac{(x+h)^3-x^3}{h}=\\\\=\lim_{h \to 0} \frac{x^3+3x^2h+3xh^2+h^2-x^3}{h}=\lim_{h \to 0} \frac{h(3x^2+3xh+h)}{h}=\\\\=\lim_{h \to 0} 3x^2+3xh+h=3x^2$