Find the derivative of f(x)=x^3 sin(x)

Find the derivative of f(x)=x³ sin(x).

To do this calculation we need to use the product rule of differentiation: if f(x)=u(x)v(x), then the derivative is f'(x)=u'(x)v(x)+u(x)v'(x). In our case, u(x)=x³ and v(x)=sin(x).

First we calculate the derivatives of u and v in the usual way:

u'(x)=3x²
v'(x)=cos(x)

Then we put together our answer using the product rule:

f'(x)= u'(x)v(x)+u(x)v'(x)
     = 3x² sin(x) + x³ cos(x)
     = x²(3 sin(x) + x cos(x))

In the final step we simplified our answer by identifying the common factor x². This step is not essential, but it is generally a good idea to simplify your answer as far as possible.