Question

resolver por la definicion de derivada

f(x) = 2 / x+3

Answer 1

Hola!

La definición de derivada se denota como:

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

Entonces, la derivada de la función es la siguiente:

$f'(x) = lim_{h \to 0} \frac{ \frac{2}{x + h + 3} - \frac{2}{x + 3}}{h} \\f'(x) = lim_{h \to 0} \frac{ \frac{2(x + 3) - 2(x + h + 3)}{(x + h + 3)(x + 3)} }{h} \\ f'(x) = lim_{h \to 0} \frac{2(x + 3) - 2(x + h + 3)}{h(x + h + 3)(x + 3)} \\ f'(x) = lim_{h \to 0} \frac{2x + 6 - 2x - 2h - 6}{h(x + h + 3)(x + 3)} \\ f'(x) = lim_{h \to 0} \frac{ - 2h}{h(x + h + 3)(x + 3)} \\ f'(x) = lim_{h \to 0} \frac{ - 2}{(x + h + 3)(x + 3)} \\ f'(x) = \frac{ - 2}{(x + 0 + 3)(x + 3)} \\ f'(x) = - \frac{2 }{(x + 3)(x + 3)} \\ \boxed{f'(x) = - \frac{2}{ {(x + 3)}^{2} } }$
Espero que te sirva, Saludos.