Problem of the Week

Updated at Aug 1, 2016 3:10 PM

This week's problem comes from the calculus category.

How would you differentiate x9sec⁡x{x}^{9}\sec{x}?

Let's begin!



ddxx9sec⁡x\frac{d}{dx} {x}^{9}\sec{x}

1
Use Product Rule to find the derivative of x9sec⁡x{x}^{9}\sec{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx9)sec⁡x+x9(ddxsec⁡x)(\frac{d}{dx} {x}^{9})\sec{x}+{x}^{9}(\frac{d}{dx} \sec{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
9x8sec⁡x+x9(ddxsec⁡x)9{x}^{8}\sec{x}+{x}^{9}(\frac{d}{dx} \sec{x})

3
Use Trigonometric Differentiation: the derivative of sec⁡x\sec{x} is sec⁡xtan⁡x\sec{x}\tan{x}.
9x8sec⁡x+x9sec⁡xtan⁡x9{x}^{8}\sec{x}+{x}^{9}\sec{x}\tan{x}

Done
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