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: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\ln{({x}^{8})}+8\]

Done
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