Question

derivada de f(x)=x^2sen3x

Answer 1

Derivaremos $f$ pela regra do produto:

$f(x)=x^{2}\mathtt{sen}(3x)\\\\f'(x)=\frac{d}{dx}[x^{2}\mathtt{sen}(3x)]\\\\f'(x)=\mathtt{sen}(3x)\frac{d}{dx}x^{2}+x^{2}\frac{d}{dx}\mathtt{sen}(3x)\\\\f'(x)=\mathtt{sen}(3x)\cdot2x+x^{2}\frac{d}{dx}\mathtt{sen}(3x)$

Deriva-se $h(x)=\mathtt{sen}(3x)$ pela regra da cadeia:

$h'(x)=\frac{d}{dx}\mathtt{sen}(3x)=\mathtt{cos}(3x)\cdot\frac{d}{dx}(3x)=\mathtt{cos}(3x)\cdot3=3\mathtt{cos}(3x)$

logo:

$f'(x)=2x\cdot\mathtt{sen}(3x)+x^{2}\cdot3\mathtt{cos}(3x)\\\\\boxed{\boxed{f'(x)=2x\cdot\mathtt{sen}(3x)+3x^{2}\cdot\mathtt{cos}(3x)}}$