Problem of the Week

Updated at Dec 22, 2014 2:18 PM

This week we have another calculus problem:

How can we find the derivative of \(x\ln{({x}^{7})}\)?

Let's start!



\[\frac{d}{dx} x\ln{({x}^{7})}\]

1
Use Product Rule to find the derivative of \(x\ln{({x}^{7})}\). The product rule states that \((fg)'=f'g+fg'\).
\[(\frac{d}{dx} x)\ln{({x}^{7})}+x(\frac{d}{dx} \ln{({x}^{7})})\]

2
Use Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\ln{({x}^{7})}+x(\frac{d}{dx} \ln{({x}^{7})})\]

3
Use Chain Rule on \(\frac{d}{dx} \ln{({x}^{7})}\). Let \(u={x}^{7}\). The derivative of \(\ln{u}\) is \(\frac{1}{u}\).
\[\ln{({x}^{7})}+\frac{x(\frac{d}{dx} {x}^{7})}{{x}^{7}}\]

4
Use Power Rule: \(\frac{d}{dx} {x}^{n}=n{x}^{n-1}\).
\[\ln{({x}^{7})}+7\]

Done
Cymath foriOSAndroid