derivate $f(x) = x\sin(3 x)\cdot e^{2 x^2}$

Question

How to derivate: $f(x) = x\sin(3 x)\cdot e^{2x^2}$? I tried to divide it in two functions, $h(x)=x\sin(3x)$ $g(x)= e^{2 x^2}$ and do $h'x+g'h= e^{2 x^2}(\sin(3 x)+3\cos(3x)+4xe^{2 x^2}(x\sin(3 x))$

Answer 1

Guide:

• You intend to compute $h'\color{red}g + g'h.$

• Note that we have $h'(x)=\sin(3x)+3\color{red}x \cos(3x)$

• Also check out for whether your brackets match.

• You might like to factor out $e^{2x^2}$ for simplification.

Answer 2

Hint

For this kind of problems, in which only products, quotients, powers, exponentials are involved, logarithmic differentiation makes life easier.

$$y=x \, \sin(3x) \, e^{2x^2} \implies \log(y)=\log(x)+\log(\sin(3x))+2x^2$$ Differentiate both sides $$\frac{y'} y=\frac 1x+\frac{3 \cos(3x)}{\sin(3x)}+4x$$ Now, use $$y'=y \times \left(\frac{y'} y\right)=x \, \sin(3x) \, e^{2x^2}\left(\frac 1x+\frac{3 \cos(3x)}{\sin(3x)}+4x \right)$$ and simplify as much as you can.