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) \]