Problem of the Week

Updated at Oct 14, 2013 8:37 AM

How would you differentiate excsc⁡x{e}^{x}\csc{x}?

Below is the solution.



ddxexcsc⁡x\frac{d}{dx} {e}^{x}\csc{x}

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

2
The derivative of ex{e}^{x} is ex{e}^{x}.
excsc⁡x+ex(ddxcsc⁡x){e}^{x}\csc{x}+{e}^{x}(\frac{d}{dx} \csc{x})

3
Use Trigonometric Differentiation: the derivative of csc⁡x\csc{x} is −csc⁡xcot⁡x-\csc{x}\cot{x}.
excsc⁡x−excsc⁡xcot⁡x{e}^{x}\csc{x}-{e}^{x}\csc{x}\cot{x}

Done
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