Problem of the Week

Updated at Jul 14, 2014 5:22 PM

How would you differentiate x7sec⁡x{x}^{7}\sec{x}?

Below is the solution.



ddxx7sec⁡x\frac{d}{dx} {x}^{7}\sec{x}

1
Use Product Rule to find the derivative of x7sec⁡x{x}^{7}\sec{x}. The product rule states that (fg)′=f′g+fg′(fg)'=f'g+fg'.
(ddxx7)sec⁡x+x7(ddxsec⁡x)(\frac{d}{dx} {x}^{7})\sec{x}+{x}^{7}(\frac{d}{dx} \sec{x})

2
Use Power Rule: ddxxn=nxn−1\frac{d}{dx} {x}^{n}=n{x}^{n-1}.
7x6sec⁡x+x7(ddxsec⁡x)7{x}^{6}\sec{x}+{x}^{7}(\frac{d}{dx} \sec{x})

3
Use Trigonometric Differentiation: the derivative of sec⁡x\sec{x} is sec⁡xtan⁡x\sec{x}\tan{x}.
7x6sec⁡x+x7sec⁡xtan⁡x7{x}^{6}\sec{x}+{x}^{7}\sec{x}\tan{x}

Done
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