(x + 1)(x + 2)2 (x + 3)3

Vamos calcular a derivada de \(f(x)=(x+1)(x+2)^{2}(x+3)^{3}\).

Façamos:

\(\\ f(x)=g(x)w(x)\\\)

Onde \(g(x)=(x+1)\\ w(x)=(x+2)^{2}(x+3)^{3}\)

Desta forma, teremos:

\(g=x+1\\\:w=\left(x+2\right)^2\left(x+3\right)^3\)

Aplicando a regra do produto:

\(\left(g\cdot w\right)'=g\:'\cdot w+g\cdot w'\)\(=\frac{d}{dx}\left(x+1\right)\left(x+2\right)^2\left(x+3\right)^3+\frac{d}{dx}\left(\left(x+2\right)^2\left(x+3\right)^3\right)\left(x+1\right)\)

\(=1\cdot \left(x+2\right)^2\left(x+3\right)^3+\left(2\left(x+2\right)\left(x+3\right)^3+3\left(x+3\right)^2\left(x+2\right)^2\right)\left(x+1\right)\)

\(=1\cdot \left(x+2\right)^2\left(x+3\right)^3+\left(2\left(x+2\right)\left(x+3\right)^3+3\left(x+3\right)^2\left(x+2\right)^2\right)\left(x+1\right)\)