Problem of the Week

Updated at Jun 26, 2017 11:10 AM

This week's problem comes from the calculus category.

How would you differentiate x4sec⁡x{x}^{4}\sec{x}?

Let's begin!



ddxx4sec⁡x\frac{d}{dx} {x}^{4}\sec{x}

1
Use Product Rule to find the derivative of x4sec⁡x{x}^{4}\sec{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx4)sec⁡x+x4(ddxsec⁡x)(\frac{d}{dx} {x}^{4})\sec{x}+{x}^{4}(\frac{d}{dx} \sec{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
4x3sec⁡x+x4(ddxsec⁡x)4{x}^{3}\sec{x}+{x}^{4}(\frac{d}{dx} \sec{x})

3
Use Trigonometric Differentiation: the derivative of sec⁡x\sec{x} is sec⁡xtan⁡x\sec{x}\tan{x}.
4x3sec⁡x+x4sec⁡xtan⁡x4{x}^{3}\sec{x}+{x}^{4}\sec{x}\tan{x}

Done
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