Problem of the Week

Updated at Dec 12, 2016 11:06 AM

For this week we've brought you this calculus problem.

How can we find the derivative of xln⁡(x4)x\ln{({x}^{4})}?

Here are the steps:



ddxxln⁡(x4)\frac{d}{dx} x\ln{({x}^{4})}

1
Use Product Rule to find the derivative of xln⁡(x4)x\ln{({x}^{4})}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx)ln⁡(x4)+x(ddxln⁡(x4))(\frac{d}{dx} x)\ln{({x}^{4})}+x(\frac{d}{dx} \ln{({x}^{4})})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
ln⁡(x4)+x(ddxln⁡(x4))\ln{({x}^{4})}+x(\frac{d}{dx} \ln{({x}^{4})})

3
Use Chain Rule on ddxln⁡(x4)\frac{d}{dx} \ln{({x}^{4})}. Let u=x4u={x}^{4}. The derivative of ln⁡u\ln{u} is 1u\frac{1}{u}.
ln⁡(x4)+x(ddxx4)x4\ln{({x}^{4})}+\frac{x(\frac{d}{dx} {x}^{4})}{{x}^{4}}

4
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
ln⁡(x4)+4\ln{({x}^{4})}+4

Done
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