Question

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

Answer 1

`d/dx(x^3lnx)=x^2 + 3x^2lnx`

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 `f(x) = x^3lnx` we have;

`{ ("Let "u = x^3, => , (du)/dx = 3x^2'), ("And "v = lnx, =>, (dv)/dx = 1/x' ) :}`

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`