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.
MM
