# Problem of the Week

## Updated at Nov 9, 2015 4:42 PM

How can we find the derivative of $$x\ln{({x}^{3})}$$?

Below is the solution.

$\frac{d}{dx} x\ln{({x}^{3})}$

 1 Use Product Rule to find the derivative of $$x\ln{({x}^{3})}$$. The product rule states that $$(fg)'=f'g+fg'$$.$(\frac{d}{dx} x)\ln{({x}^{3})}+x(\frac{d}{dx} \ln{({x}^{3})})$2 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$\ln{({x}^{3})}+x(\frac{d}{dx} \ln{({x}^{3})})$3 Use Chain Rule on $$\frac{d}{dx} \ln{({x}^{3})}$$. Let $$u={x}^{3}$$. The derivative of $$\ln{u}$$ is $$\frac{1}{u}$$.$\ln{({x}^{3})}+\frac{x(\frac{d}{dx} {x}^{3})}{{x}^{3}}$4 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$\ln{({x}^{3})}+3$Doneln(x^3)+3