Question

f(x)=(x+1). (x²+x) calcule a derivada

Answer 1

Lembrando que:

$\dfrac{d}{dx}(g(x) \cdot h(x) ) = g(x) \cdot \dfrac{d}{dx}h(x) + h(x)\cdot \dfrac{d}{dx}g(x)$

Derivada de cada fator:

$\dfrac{d}{dx}(x + 1) = \dfrac{d}{dx}x + \dfrac{d}{dx}1 = 1 \cdot x^0 + 1 = 1\\\\\dfrac{d}{dx}(x^2+x) = \dfrac{d}{dx}x^2 + \dfrac{d}{dx}x = 2x^{2-1} + x = 2x + 1$

Assim:

$\dfrac{d}{dx}(x+1)(x^2+x)=(x+1)\dfrac{d}{dx}(x^2+x) + (x^2+x) \dfrac{d}{dx}(x+1) = \\\\(x+1)(2x+1) + (x^2 + x ) \cdot 1 = 2x^2+x+2x+1+x^2+x = \boxed{3x^2+4x+1}$