Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 08

MATH100 A December 2023

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Question 08
Let $$x^{2}(f(x))^{2}=9$$ and ${\textstyle f(3)=-1}$. Find ${\textstyle f'(3)}$.

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
The expression given doesn’t have ${\textstyle f(x)}$ explicitly defined, so what sort of differentiation do you think you should use?

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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 have an expression where ${\textstyle f(x)}$ is implicitly defined, so, we need to use implicit differentiation! $${\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}x^{2}(f(x))^{2}&={\frac {\mathrm {d} }{\mathrm {d} x}}9\\2x(f(x))^{2}+x^{2}2f(x)f'(x)&=0\\f'(x)&={\frac {-2x(f(x))^{2}}{2x^{2}f(x)}}\\f'(x)&={\frac {-1}{x}}f(x)\end{aligned}}$$ Now, knowing ${\textstyle f(3)=-1}$, we can evaluate ${\textstyle f'(3)}$: ${\textstyle f'(3)={\frac {-1}{(3)}}(-1)={\frac {1}{3}}}$.