Science:Math Exam Resources/Courses/MATH110/April 2016/Question 05 (g)

MATH110 April 2016

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Question 05 (g)
Let $f(x)={\frac {x-1}{x^{2}}}$. (g) Compute $f''(x)$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Use the answer to part (d). Then use the quotient rule or break the function into a sum of functions and differentiate each of them individually.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We could use the quotient rule to calculate the derivative, but instead we will break the function into a sum of parts and differentiate them individually. From part (d), we have $f'(x)={\frac {2-x}{x^{3}}}={\frac {2}{x^{3}}}-{\frac {x}{x^{3}}}={\frac {-1}{x^{2}}}+{\frac {2}{x^{3}}}=-x^{-2}+2x^{-3}$ Then $f''(x)={\frac {d}{dx}}(-x^{-2})+{\frac {d}{dx}}(2x^{-3}).$ And using the Power Rule of differentiation gives ${\frac {d}{dx}}(-x^{-2})+{\frac {d}{dx}}(2x^{-3})=2x^{-3}-6x^{-4}={\frac {2}{x^{3}}}-{\frac {6}{x^{4}}}.$ Hence, $\color {blue}f''(x)={\frac {2}{x^{3}}}-{\frac {6}{x^{4}}}.$