# Problem of the Week

## Updated at Mar 16, 2015 9:02 AM

How can we solve for the derivative of $$x\ln{({x}^{8})}$$?

Below is the solution.

$\frac{d}{dx} x\ln{({x}^{8})}$

 1 Use Product Rule to find the derivative of $$x\ln{({x}^{8})}$$. The product rule states that $$(fg)'=f'g+fg'$$.$(\frac{d}{dx} x)\ln{({x}^{8})}+x(\frac{d}{dx} \ln{({x}^{8})})$2 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$\ln{({x}^{8})}+x(\frac{d}{dx} \ln{({x}^{8})})$3 Use Chain Rule on $$\frac{d}{dx} \ln{({x}^{8})}$$. Let $$u={x}^{8}$$. The derivative of $$\ln{u}$$ is $$\frac{1}{u}$$.$\ln{({x}^{8})}+\frac{x(\frac{d}{dx} {x}^{8})}{{x}^{8}}$4 Use Power Rule: $$\frac{d}{dx} {x}^{n}=n{x}^{n-1}$$.$\ln{({x}^{8})}+8$Doneln(x^8)+8