MATH100 December 2013
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Question 03 (a)

ShortAnswer Question. Simplify your answer as much as possible and show your work.
If a function y = ƒ(x) is defined implicitly by an equation $x^{3}+xy=5xy^{3}$, find $\displaystyle {\frac {dy}{dx}}$.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Differentiate both sides implicitly with respect to x. It is permissible to differentiate termbyterm, but don't forget that y is a function of x!

Hint 2

$\displaystyle {\frac {d}{dx}}(y^{n})=ny^{n1}\cdot {\frac {d}{dx}}(y)$
when $\displaystyle y=f(x)$.

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Solution

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Differentiating both sides implicitly with respect to x,
 ${\begin{aligned}{\frac {d}{dx}}(x^{3}+xy)&={\frac {d}{dx}}(5xy^{3})\\{\frac {d}{dx}}(x^{3})+{\color {OliveGreen}{\frac {d}{dx}}(xy)}&={\frac {d}{dx}}(5x){\color {OliveGreen}{\frac {d}{dx}}(y^{3})}\\3x^{2}+{\color {OliveGreen}1\cdot y+x\cdot {\frac {d}{dx}}(y)}&=5{\color {OliveGreen}3y^{2}\cdot {\frac {d}{dx}}(y)}\\3x^{2}+{\color {OliveGreen}y+x\cdot {\frac {dy}{dx}}}&=5{\color {OliveGreen}3y^{2}\cdot {\frac {dy}{dx}}}\end{aligned}}$
Solving for ${\frac {dy}{dx}}$ we find
 ${\begin{aligned}x\cdot {\frac {dy}{dx}}+3y^{2}\cdot {\frac {dy}{dx}}&=53x^{2}y\\{\frac {dy}{dx}}\cdot (x+3y^{2})&=53x^{2}y\\{\frac {dy}{dx}}&={\color {blue}{\frac {53x^{2}y}{x+3y^{2}}}}\end{aligned}}$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Implicit differentiation, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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