Difference between revisions of "022 Sample Final A, Problem 4"

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|When we use implicit differentiation, we combine the chain rule with the fact that <math style="vertical-align: -18%">y</math> is a function of <math style="vertical-align: 0%">x</math>, and could really be written as <math style="vertical-align: -21%">y(x).</math> Because of this, the derivative of <math style="vertical-align:-17%">y^3</math> with respect to <math style="vertical-align: 0%">x</math> requires the chain rule, so  
 
|When we use implicit differentiation, we combine the chain rule with the fact that <math style="vertical-align: -18%">y</math> is a function of <math style="vertical-align: 0%">x</math>, and could really be written as <math style="vertical-align: -21%">y(x).</math> Because of this, the derivative of <math style="vertical-align:-17%">y^3</math> with respect to <math style="vertical-align: 0%">x</math> requires the chain rule, so  
 
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|&nbsp;&nbsp;&nbsp;&nbsp; <math>\frac{d}{dx}\left(y^{3}\right)=3y^{2}\cdot\frac{dy}{dx}.</math>
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::<math>\frac{d}{dx}\left(y^{3}\right)=3y^{2}\cdot\frac{dy}{dx}.</math>
 
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|For this problem we also need to use the product rule.
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|For this problem, we also need to use the product rule.
 
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!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
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|First, we differentiate each term separately with respect to <math style="vertical-align: 0%">x</math> and apply the product rule on the right hand side to find that&thinsp; <math style="vertical-align: -18%">x + y = x^3y^3</math> &thinsp;differentiates implicitly to
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|First, we differentiate each term separately with respect to <math style="vertical-align: 0px">x</math> and apply the product rule on the right hand side to find that&thinsp; <math style="vertical-align: -4px">x + y = x^3y^3</math> &thinsp;differentiates implicitly to
 
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::<math>1 + \frac{dy}{dx} = 3x^2y^3 + 3x^3y^2 \cdot \frac{dy}{dx}</math>.
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::<math>1 + \frac{dy}{dx} \,=\, 3x^2y^3 + 3x^3y^2 \cdot \frac{dy}{dx}</math>.
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
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|Now we need to solve for <math>\frac{dy}{dx}</math> and doing so we find that <math>\frac{dy}{dx} = \frac{3x^2y^3 - 1}{1 - 3x^3y^2}</math>
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|Now we need to solve for &thinsp;<math style="vertical-align: -14px">\frac{dy}{dx}</math>&thinsp;, and doing so we find that &nbsp;<math style="vertical-align: -18px">\frac{dy}{dx} \,=\, \frac{3x^2y^3 - 1}{1 - 3x^3y^2}</math>.
 
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!Final Answer: &nbsp;
 
!Final Answer: &nbsp;
 
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|<math>\frac{dy}{dx} = \frac{3x^2y^3 - 1}{1 - 3x^3y^2}</math>
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::<math>\frac{dy}{dx} \,=\, \frac{3x^2y^3 - 1}{1 - 3x^3y^2}.</math>
 
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[[022_Sample Final A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Sample Final A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 17:33, 6 June 2015


Use implicit differentiation to find  

Foundations:  
When we use implicit differentiation, we combine the chain rule with the fact that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , and could really be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x).} Because of this, the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^3} with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} requires the chain rule, so
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\left(y^{3}\right)=3y^{2}\cdot\frac{dy}{dx}.}
For this problem, we also need to use the product rule.

Solution:

Step 1:  
First, we differentiate each term separately with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and apply the product rule on the right hand side to find that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + y = x^3y^3}  differentiates implicitly to
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 + \frac{dy}{dx} \,=\, 3x^2y^3 + 3x^3y^2 \cdot \frac{dy}{dx}} .
Step 2:  
Now we need to solve for  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}}  , and doing so we find that  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx} \,=\, \frac{3x^2y^3 - 1}{1 - 3x^3y^2}} .
Final Answer:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx} \,=\, \frac{3x^2y^3 - 1}{1 - 3x^3y^2}.}

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