Calculus: derivative of products of functions of x.

Q. What is the derivative of $x^2 e^x \sin(x)$ ?


A. The derivative of a product of three functions in $x$, say, $u, v$ and $w$,$\frac{d(uvw)}{d x}$ is given by

$$\frac{d(uvw)}{dx} =\frac{uvw}{u} \frac{dw}{dx} + \frac{uvw}{v} \frac{dw}{dv} + \frac{uvw}{u} \frac{dw}{dv}$$ .

It is easy to generalize this 'product' rule. Now let $u = x^2, v = e^x, w = \sin{(x)}$. Then we obtain the following result:

$$e^x \sin(x) \frac{d(x^2)}{dx} + x^2 \sin(x) \frac{d(e^x)}{dx}+ {x^2 e^x}\frac{d(\sin(x) )}{dx} = 2x e^x \sin(x) + x^2 \sin(x) e^x + x^2 e^x \cos(x)$$