Science:Math Exam Resources/Courses/MATH110/December 2016/Question 08 (b)

MATH110 December 2016

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Question 08 (b)
Find $y'$ if $y=t^{2}e^{(t^{2})}$.

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 product rule and the chain rule.

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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. Note that the bracket in the exponent ${\textstyle (t^{2})}$ is redundant: ${\textstyle e^{t^{2}}}$ means ${\textstyle e^{(t^{2})}}$. Let us write ${\textstyle f(t)=te^{t}}$ and ${\textstyle g(t)=t^{2}}$. Then ${\textstyle y=t^{2}e^{t^{2}}=f(g(t))}$. Using the product rule with ${\textstyle F(t)=t}$, ${\textstyle G(t)=e^{t}}$, ${\textstyle F'(t)=1}$, ${\textstyle G'(t)=e^{t}}$, the derivatives of ${\textstyle f(t)=F(t)G(t)}$ and ${\textstyle g(t)}$ are given by $$f'(t)=F'(t)G(t)+F(t)G'(t)=1\cdot e^{t}+te^{t}=(t+1)e^{t},$$ $$g'(t)=2t.$$ Now, by the chain rule, $${\begin{aligned}y'(t)&=f'(g(t))g'(t)\\&=(t^{2}+1)e^{t^{2}}\cdot 2t\\&=2t(t^{2}+1)e^{t^{2}}.\end{aligned}}$$ Answer: ${\textstyle y'(t)={\color {blue}2t(t^{2}+1)e^{t^{2}}}}$.