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⁡xex\ln{x}{e}^{x}?

Here are the steps:



ddxln⁡xex\frac{d}{dx} \ln{x}{e}^{x}

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

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

3
The derivative of ex{e}^{x} is ex{e}^{x}.
exx+ln⁡xex\frac{{e}^{x}}{x}+\ln{x}{e}^{x}

Done
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