Problem of the Week

Updated at Mar 16, 2015 9:02 AM

How can we solve for the derivative of xln⁡(x8)x\ln{({x}^{8})}?

Below is the solution.



ddxxln⁡(x8)\frac{d}{dx} x\ln{({x}^{8})}

1
Use Product Rule to find the derivative of xln⁡(x8)x\ln{({x}^{8})}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx)ln⁡(x8)+x(ddxln⁡(x8))(\frac{d}{dx} x)\ln{({x}^{8})}+x(\frac{d}{dx} \ln{({x}^{8})})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
ln⁡(x8)+x(ddxln⁡(x8))\ln{({x}^{8})}+x(\frac{d}{dx} \ln{({x}^{8})})

3
Use Chain Rule on ddxln⁡(x8)\frac{d}{dx} \ln{({x}^{8})}. Let u=x8u={x}^{8}. The derivative of ln⁡u\ln{u} is 1u\frac{1}{u}.
ln⁡(x8)+x(ddxx8)x8\ln{({x}^{8})}+\frac{x(\frac{d}{dx} {x}^{8})}{{x}^{8}}

4
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
ln⁡(x8)+8\ln{({x}^{8})}+8

Done
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