Science:Math Exam Resources/Courses/MATH110/December 2012/Question 03 (a)

MATH110 December 2012

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Question 03 (a)
Calculate the following derivative. You may leave your answer unsimplified. $\displaystyle f'(9)$, where $\displaystyle f(x)=g(x)x^{1/2}$, $\displaystyle g(9)=2$, and $\displaystyle g'(9)=4$.

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
To take the derivative of $f(x)$ you'll need the product rule. Remember that the derivative of $g(x)$ is $g'(x)$.

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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 apply the product rule to $f(x)$ to get: $\displaystyle f'(x)=g'(x)*x^{1/2}+g(x)*(1/2)(x^{-1/2})$ Now we simply evaluate at x = 9, using the other information given in the problem to fill in g(9) and g'(9). Recall also that $x^{1/2}={\sqrt {x}}$. ${\begin{aligned}f'(9)&=g'(9)*(9)^{1/2}+g(9)*(1/2)(9^{-1/2})\\&=4*3+2*(1/2)(1/3)\\&=12+1/3\end{aligned}}$ So the value of $\displaystyle f'(9)$ is $37/3$.

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