Calcular la derivada por definición:
$f(x)=5x^2-2x+6$
Solución
| Definición de derivada | ||
| $f'(x)=\lim \limits_{h \to 0}{\dfrac{f(x+h)-f(x)}{h}}$ | ||
| Calculando $f(x+h)$ | ||
| $f(x+h)=5(x+h)^2 -2(x+h)+6$ | ||
| $f(x+h)=5(x^2+2xh+h^2)-2x-2h+6$ | ||
| $f(x+h)=5x^2+10xh+5h^2-2x-2h+6$ | ||
| Reemplazando: | ||
| $f'(x)=\lim \limits_{h \to 0}{\dfrac{f(x+h)-f(x)}{h}}$ | ||
| $f'(x)=\lim \limits_{h \to 0}{\dfrac{5x^2+10xh+5h^2-2x-2h+6-(5x^2-2x+6)}{h}}$ | ||
| $f'(x)=\lim \limits_{h \to 0}{\dfrac{5x^2+10xh+5h^2-2x-2h+6-5x^2+2x-6}{h}}$ | ||
| $f'(x)=\lim \limits_{h \to 0}{\dfrac{10xh+5h^2-2h}{h}}$ | ||
| $f'(x)=\lim \limits_{h \to 0}{(10x+5h-2)}$ | ||
| Reemplazando $h=0$: | ||
| $f'(x)=10x+5(0)-2$ | ||
| $f'(x)=10x-2$ | ||
| Respuesta: | ||
| $f'(x)=10x-2$ | ||