# How do you differentiate #f(x)=x^3lnx # using the product rule?

##### 1 Answer

Dec 12, 2016

#### Answer:

#### Explanation:

If you are studying maths, then you should learn the Product Rule for Differentiation, and practice how to use it:

# d/dx(uv)=u(dv)/dx+(du)/dxv # , or,# (uv)' = (du)v + u(dv) #

I was taught to remember the rule in words; "*The first times the derivative of the second plus the derivative of the first times the second* ".

This can be extended to three products:

# d/dx(uvw)=uv(dw)/dx+u(dv)/dxw + (du)/dxvw#

So with

Applying the product rule we get:

# \ \ \ \ \ \ \ \ \ \ \ d/dx(uv)=u(dv)/dx + (du)/dxv #

# :. d/dx(x^3lnx)=(x^3)(1/x) + (3x^2)(lnx) #

# :. d/dx(x^3lnx)=x^2 + 3x^2lnx #