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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #461 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yxx}\\&\text{Where }f(x,y,z)=-5ln{(y^4)}e^{(z)}sin{(x)}\end{align*}$

Possible Answers:

${\frac{-{(20e^{(z)}cos{(x)})}}{y}}$

${\frac{{(20e^{(z)}sin{(x)})}}{y}}$

$\frac{-{(500ln{(y^4)}^2e^{(3z)}cos{(x)}^2sin{(x)})}}{y}$

${-\frac{ {(20e^{(z)}sin{(x)})}}{y} - 10ln{(y^4)}e^{(z)}cos{(x)}}$

Correct answer:

${\frac{{(20e^{(z)}sin{(x)})}}{y}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&f_{y}=\frac{-{(20e^{(z)}sin{(x)})}}{y}\\&f_{yx}=\frac{-{(20e^{(z)}cos{(x)})}}{y}\\&f_{yxx}=\frac{{(20e^{(z)}sin{(x)})}}{y}\end{align*}$

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Example Question #462 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yy}\\&\text{Where }f(x,y,z)=-6cos{(z^2)}ln{(y^2)}\end{align*}$

Possible Answers:

${\frac{-{(24cos{(z^2)})}}{y}}$

${\frac{{(12cos{(z^2)})}}{y^2}}$

$\frac{{(144cos{(z^2)}^2)}}{y^2}$

${\frac{{(12cos{(z^2)})}}{y^2} -\frac{ {(12cos{(z^2)})}}{y}}$

Correct answer:

${\frac{{(12cos{(z^2)})}}{y^2}}$

Explanation:

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Example Question #463 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{z}\\&\text{Where }f(x,y,z)=-3\cdot2^{(z^2)}\cdot3^{(x^3)}ln{(y^3)}\end{align*}$

Possible Answers:

${-4.159\cdot2^{(z^2)}\cdot3^{(x^3)}zln{(y^3)}}$

$-4.159\cdot2^{(z^2)}\cdot3^{(x^3)}zln{(y^3)}$

${-4.159\cdot2^{(z^2)}\cdot3^{(x^3)}zln{(y^3)}}$

Correct answer:

${-4.159\cdot2^{(z^2)}\cdot3^{(x^3)}zln{(y^3)}}$

Explanation:

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Example Question #464 : Partial Derivatives

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

Possible Answers:

${4z^3e^{(y)}sin{(x)}}$

${z^4e^{(y)}sin{(x)} - 4z^3e^{(y)}cos{(x)} - z^4e^{(y)}cos{(x)}}$

${4z^3e^{(y)}sin{(x)} + 2z^4e^{(y)}sin{(x)}}$

$4z^11e^{(3y)}cos{(x)}^2sin{(x)}$

Correct answer:

${4z^3e^{(y)}sin{(x)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[cos(u)]=-sin(u)du\\&d[e^u]=e^udu\\&f_{x}=z^4e^{(y)}sin{(x)}\\&f_{xy}=z^4e^{(y)}sin{(x)}\\&f_{xyz}=4z^3e^{(y)}sin{(x)}\end{align*}$

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Example Question #465 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zz}\\&\text{Where }f(x,y,z)=-3\cdot2^{(x^4)}y^3e^{(z^2)}\end{align*}$

Possible Answers:

$36\cdot2^{(2x^4)}y^6z^2e^{(2z^2)}$

${- 6\cdot2^{(x^4)}y^3e^{(z^2)} - 6\cdot2^{(x^4)}y^3ze^{(z^2)} - 12\cdot2^{(x^4)}y^3z^2e^{(z^2)}}$

${-12\cdot2^{(x^4)}y^3ze^{(z^2)}}$

${- 6\cdot2^{(x^4)}y^3e^{(z^2)} - 12\cdot2^{(x^4)}y^3z^2e^{(z^2)}}$

Correct answer:

${- 6\cdot2^{(x^4)}y^3e^{(z^2)} - 12\cdot2^{(x^4)}y^3z^2e^{(z^2)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[e^u]=e^udu\\&f_{z}=-6\cdot2^{(x^4)}y^3ze^{(z^2)}\\&f_{zz}=- 6\cdot2^{(x^4)}y^3e^{(z^2)} - 12\cdot2^{(x^4)}y^3z^2e^{(z^2)}\end{align*}$

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Example Question #466 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zy}\\&\text{Where }f(x,y,z)=-5cos{(y^2)}cos{(z^4)}sin{(x^4)}\end{align*}$

Possible Answers:

$200yz^3cos{(y^2)}cos{(z^4)}sin{(x^4)}^2sin{(y^2)}sin{(z^4)}$

${20z^3cos{(y^2)}sin{(x^4)}sin{(z^4)} + 10ycos{(z^4)}sin{(x^4)}sin{(y^2)}}$

${-40yz^3sin{(x^4)}sin{(y^2)}sin{(z^4)}}$

${20z^3cos{(y^2)}sin{(x^4)}sin{(z^4)} - 40yz^3sin{(x^4)}sin{(y^2)}sin{(z^4)}}$

Correct answer:

${-40yz^3sin{(x^4)}sin{(y^2)}sin{(z^4)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[cos(u)]=-sin(u)du\\&f_{z}=20z^3cos{(y^2)}sin{(x^4)}sin{(z^4)}\\&f_{zy}=-40yz^3sin{(x^4)}sin{(y^2)}sin{(z^4)}\end{align*}$

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Example Question #467 : Partial Derivatives

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

Possible Answers:

${6xcos{(z^4)}ln{(y^3)} - 12x^2z^3ln{(y^3)}sin{(z^4)}}$

$-72x^3z^3cos{(z^4)}ln{(y^3)}^2sin{(z^4)}$

${6xcos{(z^4)}ln{(y^3)} - 24xz^3ln{(y^3)}sin{(z^4)}}$

${-24xz^3ln{(y^3)}sin{(z^4)}}$

Correct answer:

${-24xz^3ln{(y^3)}sin{(z^4)}}$

Explanation:

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Example Question #468 : Partial Derivatives

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

Possible Answers:

${- 18sin{(y^2)}sin{(z^4)} - 18xsin{(y^2)}sin{(z^4)} - 72z^3cos{(z^4)}sin{(y^2)} - 144yz^3cos{(y^2)}cos{(z^4)}}$

${- 36xsin{(y^2)}sin{(z^4)} - 18x^2ycos{(y^2)}sin{(z^4)} - 36x^2z^3cos{(z^4)}sin{(y^2)}}$

${-144yz^3cos{(y^2)}cos{(z^4)}}$

$209952x^6yz^3cos{(y^2)}cos{(z^4)}sin{(y^2)}^3sin{(z^4)}^3$

Correct answer:

${-144yz^3cos{(y^2)}cos{(z^4)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[sin(u)]=cos(u)du\\&f_{x}=-18xsin{(y^2)}sin{(z^4)}\\&f_{xx}=-18sin{(y^2)}sin{(z^4)}\\&f_{xxz}=-72z^3cos{(z^4)}sin{(y^2)}\\&f_{xxzy}=-144yz^3cos{(y^2)}cos{(z^4)}\end{align*}$

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Example Question #469 : Partial Derivatives

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

Possible Answers:

$\frac{-{(128ln{(y^4)}ln{(z^2)}^3cos{(x)}sin{(x)}^2)}}{y^2}$

${-\frac{ {(16ln{(z^2)}sin{(x)})}}{y} - 2ln{(y^4)}ln{(z^2)}cos{(x)}}$

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

${\frac{{(8ln{(z^2)}cos{(x)})}}{y^2}}$

Correct answer:

${\frac{{(8ln{(z^2)}cos{(x)})}}{y^2}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&f_{y}=\frac{-{(8ln{(z^2)}sin{(x)})}}{y}\\&f_{yy}=\frac{{(8ln{(z^2)}sin{(x)})}}{y^2}\\&f_{yyx}=\frac{{(8ln{(z^2)}cos{(x)})}}{y^2}\end{align*}$

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Example Question #470 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyxz}\\&\text{Where }f(x,y,z)=-2\cdot2^{(x^4)}ln{(y^3)}ln{(z)}\end{align*}$

Possible Answers:

${\frac{{(24\cdot2^{(x^4)}x^3ln{(2)})}}{{(yz^2)}}}$

${\frac{{(24\cdot2^{(x^4)}x^3ln{(2)})}}{{(yz^2)}} -\frac{ {(2\cdot2^{(x^4)}ln{(y^3)})}}{z} -\frac{ {(24\cdot2^{(x^4)}x^3ln{(2)})}}{{(yz)}} -\frac{ {(6\cdot2^{(x^4)})}}{{(yz)}}}$

$\frac{{(192\cdot2^{(4x^4)}x^3ln{(y^3)}^3ln{(2)}ln{(z)}^2)}}{{(yz^2)}}$

${-\frac{ {(6\cdot2^{(x^4)}ln{(z)})}}{y} -\frac{ {(4\cdot2^{(x^4)}ln{(y^3)})}}{z} - 8\cdot2^{(x^4)}x^3ln{(y^3)}ln{(2)}ln{(z)}}$

Correct answer:

${\frac{{(24\cdot2^{(x^4)}x^3ln{(2)})}}{{(yz^2)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&d[ln(u)]=\frac{du}{u}\\&d[a^u]=a^uduln(a)\\&f_{z}=\frac{-{(2\cdot2^{(x^4)}ln{(y^3)})}}{z}\\&f_{zy}=\frac{-{(6\cdot2^{(x^4)})}}{{(yz)}}\\&f_{zyx}=\frac{-{(24\cdot2^{(x^4)}x^3ln{(2)})}}{{(yz)}}\\&f_{zyxz}=\frac{{(24\cdot2^{(x^4)}x^3ln{(2)})}}{{(yz^2)}}\end{align*}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #461 : Partial Derivatives

Example Question #462 : Partial Derivatives

Example Question #463 : Partial Derivatives

Example Question #464 : Partial Derivatives

Example Question #465 : Partial Derivatives

Example Question #466 : Partial Derivatives

Example Question #467 : Partial Derivatives

Example Question #468 : Partial Derivatives

Example Question #469 : Partial Derivatives

Example Question #470 : Partial Derivatives

All Calculus 3 Resources

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