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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #531 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xzyy}\\&\text{Where }f(x,y,z)=\frac{{(z^2e^{(x^2)})}}{{(5y^2)}}\end{align*}$

Possible Answers:

${\frac{{(16xz^7e^{(4x^2)})}}{{(625y^10)}}}$

${\frac{{(2ze^{(x^2)})}}{{(5y^2)}} -\frac{ {(4z^2e^{(x^2)})}}{{(5y^3)}} +\frac{ {(2xz^2e^{(x^2)})}}{{(5y^2)}}}$

${\frac{-{(1536x^4z^5e^{(4x^2)})}}{{(625y^11)}}}$

${\frac{{(24xze^{(x^2)})}}{{(5y^4)}}}$

Correct answer:

${\frac{{(24xze^{(x^2)})}}{{(5y^4)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f_{x}=\frac{{(2xz^2e^{(x^2)})}}{{(5y^2)}}\\&f_{xz}=\frac{{(4xze^{(x^2)})}}{{(5y^2)}}\\&f_{xzy}=\frac{-{(8xze^{(x^2)})}}{{(5y^3)}}\\&f_{xzyy}=\frac{{(24xze^{(x^2)})}}{{(5y^4)}}\end{align*}$

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Example Question #532 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yyy}\\&\text{Where }f(x,y,z)=-3^{(z^2)}ln{(y^2)}e^{(x^2)}\end{align*}$

Possible Answers:

${\frac{-{(8\cdot3^{(3z^2)}e^{(3x^2)})}}{y^3}}$

${\frac{-{(4\cdot3^{(z^2)}e^{(x^2)})}}{y^3}}$

${\frac{-{(6\cdot3^{(z^2)}e^{(x^2)})}}{y}}$

${\frac{{(16\cdot3^{(3z^2)}e^{(3x^2)})}}{y^6}}$

Correct answer:

${\frac{-{(4\cdot3^{(z^2)}e^{(x^2)})}}{y^3}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f_{y}=\frac{-{(2\cdot3^{(z^2)}e^{(x^2)})}}{y}\\&f_{yy}=\frac{{(2\cdot3^{(z^2)}e^{(x^2)})}}{y^2}\\&f_{yyy}=\frac{-{(4\cdot3^{(z^2)}e^{(x^2)})}}{y^3}\end{align*}$

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Example Question #533 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xxxy}\\&\text{Where }f(x,y,z)=8z^2ln{(3y)}sin{(4x)}\end{align*}$

Possible Answers:

${\frac{-{(1073741824z^8cos{(4x)}^3ln{(3y)}^3sin{(4x)})}}{y}}$

${\frac{{(8z^2sin{(4x)})}}{y} + 96z^2cos{(4x)}ln{(3y)}}$

${\frac{-{(512z^2cos{(4x)})}}{y}}$

${\frac{{(262144z^8cos{(4x)}^3ln{(3y)}^3sin{(4x)})}}{y}}$

Correct answer:

${\frac{-{(512z^2cos{(4x)})}}{y}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=8z^2ln{(3y)}sin{(4x)}\\&f_{x}=32z^2cos{(4x)}ln{(3y)}\\&f_{xx}=-128z^2ln{(3y)}sin{(4x)}\\&f_{xxx}=-512z^2cos{(4x)}ln{(3y)}\\&f_{xxxy}=\frac{-{(512z^2cos{(4x)})}}{y}\end{align*}$

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Example Question #534 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yxyz}\\&\text{Where }f(x,y,z)=\frac{-{(x^2e^{(4z)})}}{{(9y)}}\end{align*}$

Possible Answers:

${\frac{{(8x^7e^{(16z)})}}{{(6561y^6)}}}$

${\frac{{(2x^2e^{(4z)})}}{{(9y^2)}} -\frac{ {(4x^2e^{(4z)})}}{{(9y)}} -\frac{ {(2xe^{(4z)})}}{{(9y)}}}$

${\frac{-{(16xe^{(4z)})}}{{(9y^3)}}}$

${\frac{{(128x^5e^{(16z)})}}{{(6561y^10)}}}$

Correct answer:

${\frac{-{(16xe^{(4z)})}}{{(9y^3)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{-{(x^2e^{(4z)})}}{{(9y)}}\\&f_{y}=\frac{{(x^2e^{(4z)})}}{{(9y^2)}}\\&f_{yx}=\frac{{(2xe^{(4z)})}}{{(9y^2)}}\\&f_{yxy}=\frac{-{(4xe^{(4z)})}}{{(9y^3)}}\\&f_{yxyz}=\frac{-{(16xe^{(4z)})}}{{(9y^3)}}\end{align*}$

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Example Question #535 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yyx}\\&\text{Where }f(x,y,z)=\frac{-{(4^zsin{(3x)})}}{{(2y)}}\end{align*}$

Possible Answers:

${\frac{-{(3\cdot4^zcos{(3x)})}}{y^3}}$

${\frac{{(4^zsin{(3x)})}}{y^2} -\frac{ {(3\cdot4^zcos{(3x)})}}{{(2y)}}}$

${\frac{{(3\cdot4^{(3z)}cos{(3x)}sin{(3x)}^2)}}{{(2y^8)}}}$

${\frac{-{(3\cdot4^{(3z)}cos{(3x)}sin{(3x)}^2)}}{{(8y^5)}}}$

Correct answer:

${\frac{-{(3\cdot4^zcos{(3x)})}}{y^3}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{-{(4^zsin{(3x)})}}{{(2y)}}\\&f_{y}=\frac{{(4^zsin{(3x)})}}{{(2y^2)}}\\&f_{yy}=\frac{-{(4^zsin{(3x)})}}{y^3}\\&f_{yyx}=\frac{-{(3\cdot4^zcos{(3x)})}}{y^3}\end{align*}$

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Example Question #536 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xyz}\\&\text{Where }f(x,y,z)=7cos{(x^2)}e^{(z)}ln{(y)}\end{align*}$

Possible Answers:

${\frac{{(7cos{(x^2)}e^{(z)})}}{y} + 7cos{(x^2)}e^{(z)}ln{(y)} - 14xsin{(x^2)}e^{(z)}ln{(y)}}$

${\frac{-{(2744x^3sin{(x^2)}^3e^{(3z)}ln{(y)})}}{y^2}}$

${\frac{-{(14xsin{(x^2)}e^{(z)})}}{y}}$

${\frac{-{(686xcos{(x^2)}^2sin{(x^2)}e^{(3z)}ln{(y)}^2)}}{y}}$

Correct answer:

${\frac{-{(14xsin{(x^2)}e^{(z)})}}{y}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[ln(u)]=\frac{du}{u}\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=7cos{(x^2)}e^{(z)}ln{(y)}\\&f_{x}=-14xsin{(x^2)}e^{(z)}ln{(y)}\\&f_{xy}=\frac{-{(14xsin{(x^2)}e^{(z)})}}{y}\\&f_{xyz}=\frac{-{(14xsin{(x^2)}e^{(z)})}}{y}\end{align*}$

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Example Question #537 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zzx}\\&\text{Where }f(x,y,z)=2^{(y^2)}z^2e^{(x^2)}\end{align*}$

Possible Answers:

${4\cdot2^{(y^2)}xe^{(x^2)}}$

${8\cdot2^{(3y^2)}xz^4e^{(3x^2)}}$

${16\cdot2^{(3y^2)}xze^{(3x^2)}}$

${4\cdot2^{(y^2)}ze^{(x^2)} + 2\cdot2^{(y^2)}xz^2e^{(x^2)}}$

Correct answer:

${4\cdot2^{(y^2)}xe^{(x^2)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=2^{(y^2)}z^2e^{(x^2)}\\&f_{z}=2\cdot2^{(y^2)}ze^{(x^2)}\\&f_{zz}=2\cdot2^{(y^2)}e^{(x^2)}\\&f_{zzx}=4\cdot2^{(y^2)}xe^{(x^2)}\end{align*}$

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Example Question #538 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yzy}\\&\text{Where }f(x,y,z)=\frac{-{(9sin{(4x)}e^{(y^2)})}}{z}\end{align*}$

Possible Answers:

${\frac{{(2916y^2sin{(4x)}^3e^{(3y^2)})}}{z^4}}$

${\frac{{(18sin{(4x)}e^{(y^2)})}}{z^2} +\frac{ {(36y^2sin{(4x)}e^{(y^2)})}}{z^2}}$

${\frac{{(9sin{(4x)}e^{(y^2)})}}{z^2} -\frac{ {(36ysin{(4x)}e^{(y^2)})}}{z}}$

${\frac{{(9sin{(4x)}e^{(y^2)})}}{z^2} +\frac{ {(36ysin{(4x)}e^{(y^2)})}}{z}}$

Correct answer:

${\frac{{(18sin{(4x)}e^{(y^2)})}}{z^2} +\frac{ {(36y^2sin{(4x)}e^{(y^2)})}}{z^2}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{-{(9sin{(4x)}e^{(y^2)})}}{z}\\&f_{y}=\frac{-{(18ysin{(4x)}e^{(y^2)})}}{z}\\&f_{yz}=\frac{{(18ysin{(4x)}e^{(y^2)})}}{z^2}\\&f_{yzy}=\frac{{(18sin{(4x)}e^{(y^2)})}}{z^2} +\frac{ {(36y^2sin{(4x)}e^{(y^2)})}}{z^2}\end{align*}$

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Example Question #539 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyy}\\&\text{Where }f(x,y,z)=\frac{-{(3^xcos{(z^2)}cos{(y)})}}{10}\end{align*}$

Possible Answers:

${\frac{{(3^{(3x)}z^3sin{(z^2)}^3cos{(y)}^2sin{(y)})}}{125}}$

${\frac{{(3^{(3x)}zcos{(z^2)}^2sin{(z^2)}cos{(y)}sin{(y)}^2)}}{500}}$

${\frac{-{(3^xzsin{(z^2)}cos{(y)})}}{5}}$

${\frac{{(3^xcos{(z^2)}sin{(y)})}}{5} +\frac{ {(3^xzsin{(z^2)}cos{(y)})}}{5}}$

Correct answer:

${\frac{-{(3^xzsin{(z^2)}cos{(y)})}}{5}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{-{(3^xcos{(z^2)}cos{(y)})}}{10}\\&f_{z}=\frac{{(3^xzsin{(z^2)}cos{(y)})}}{5}\\&f_{zy}=\frac{-{(3^xzsin{(z^2)}sin{(y)})}}{5}\\&f_{zyy}=\frac{-{(3^xzsin{(z^2)}cos{(y)})}}{5}\end{align*}$

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Example Question #540 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yz}\\&\text{Where }f(x,y,z)=\frac{{(ln{(z^2)}e^{(x)})}}{{(7y^2)}}\end{align*}$

Possible Answers:

${\frac{{(8ln{(z^2)}e^{(2x)})}}{{(49y^6z)}}}$

${\frac{-{(4e^{(x)})}}{{(7y^3z)}}}$

${\frac{{(2e^{(x)})}}{{(7y^2z)}} -\frac{ {(2ln{(z^2)}e^{(x)})}}{{(7y^3)}}}$

${\frac{-{(4ln{(z^2)}e^{(2x)})}}{{(49y^5z)}}}$

Correct answer:

${\frac{-{(4e^{(x)})}}{{(7y^3z)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{{(ln{(z^2)}e^{(x)})}}{{(7y^2)}}\\&f_{y}=\frac{-{(2ln{(z^2)}e^{(x)})}}{{(7y^3)}}\\&f_{yz}=\frac{-{(4e^{(x)})}}{{(7y^3z)}}\end{align*}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #531 : Partial Derivatives

Example Question #532 : Partial Derivatives

Example Question #533 : Partial Derivatives

Example Question #534 : Partial Derivatives

Example Question #535 : Partial Derivatives

Example Question #536 : Partial Derivatives

Example Question #537 : Partial Derivatives

Example Question #538 : Partial Derivatives

Example Question #539 : Partial Derivatives

Example Question #540 : Partial Derivatives

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