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: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\ln{({x}^{3})}+3\]

Done
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