Problem of the Week

Updated at Mar 24, 2014 9:26 AM

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

How can we solve for the derivative of \(\ln{x}{e}^{x}\)?

Here are the steps:



\[\frac{d}{dx} \ln{x}{e}^{x}\]

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

2
The derivative of \(\ln{x}\) is \(\frac{1}{x}\).
\[\frac{{e}^{x}}{x}+\ln{x}(\frac{d}{dx} {e}^{x})\]

3
The derivative of \({e}^{x}\) is \({e}^{x}\).
\[\frac{{e}^{x}}{x}+\ln{x}{e}^{x}\]

Done
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