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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #511 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zzyx}\\&\text{Where }f(x,y,z)=-7cos{(y^3)}sin{(z^2)}sin{(x)}\end{align*}$

Possible Answers:

$-28812y^2z^2cos{(y^3)}^3cos{(z^2)}^2sin{(y^3)}sin{(z^2)}^2cos{(x)}sin{(x)}^3$

${14zcos{(y^3)}cos{(z^2)}sin{(x)}\cdot{(42y^2cos{(z^2)}sin{(y^3)}cos{(x)} - 84y^2z^2sin{(y^3)}sin{(z^2)}cos{(x)})}\cdot{(42y^2cos{(z^2)}sin{(y^3)}sin{(x)} - 84y^2z^2sin{(y^3)}sin{(z^2)}sin{(x)})}\cdot{(14cos{(y^3)}cos{(z^2)}sin{(x)} - 28z^2cos{(y^3)}sin{(z^2)}sin{(x)})}}$

${42y^2cos{(z^2)}sin{(y^3)}cos{(x)} - 84y^2z^2sin{(y^3)}sin{(z^2)}cos{(x)}}$

${21y^2sin{(y^3)}sin{(z^2)}sin{(x)} - 28zcos{(y^3)}cos{(z^2)}sin{(x)} - 7cos{(y^3)}sin{(z^2)}cos{(x)}}$

Correct answer:

${42y^2cos{(z^2)}sin{(y^3)}cos{(x)} - 84y^2z^2sin{(y^3)}sin{(z^2)}cos{(x)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f_{z}=-14zcos{(y^3)}cos{(z^2)}sin{(x)}\\&f_{zz}=28z^2cos{(y^3)}sin{(z^2)}sin{(x)} - 14cos{(y^3)}cos{(z^2)}sin{(x)}\\&f_{zzy}=42y^2cos{(z^2)}sin{(y^3)}sin{(x)} - 84y^2z^2sin{(y^3)}sin{(z^2)}sin{(x)}\\&f_{zzyx}=42y^2cos{(z^2)}sin{(y^3)}cos{(x)} - 84y^2z^2sin{(y^3)}sin{(z^2)}cos{(x)}\end{align*}$

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Example Question #512 : Partial Derivatives

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

Possible Answers:

${72x^2yz^2cos{(x^3)}e^{(y^3)} + 108x^2y^4z^2cos{(x^3)}e^{(y^3)}}$

${12x^2z^2cos{(x^3)}e^{(y^3)} + 24y^2z^2sin{(x^3)}e^{(y^3)}}$

$1728x^2y^4z^6cos{(x^3)}sin{(x^3)}^2e^{(3y^3)}$

${12y^2z^2sin{(x^3)}e^{(y^3)}\cdot{(24yz^2sin{(x^3)}e^{(y^3)} + 36y^4z^2sin{(x^3)}e^{(y^3)})}\cdot{(72x^2yz^2cos{(x^3)}e^{(y^3)} + 108x^2y^4z^2cos{(x^3)}e^{(y^3)})}}$

Correct answer:

${72x^2yz^2cos{(x^3)}e^{(y^3)} + 108x^2y^4z^2cos{(x^3)}e^{(y^3)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f_{y}=12y^2z^2sin{(x^3)}e^{(y^3)}\\&f_{yy}=24yz^2sin{(x^3)}e^{(y^3)} + 36y^4z^2sin{(x^3)}e^{(y^3)}\\&f_{yyx}=72x^2yz^2cos{(x^3)}e^{(y^3)} + 108x^2y^4z^2cos{(x^3)}e^{(y^3)}\end{align*}$

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Example Question #512 : Partial Derivatives

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

Possible Answers:

${\frac{{(288\cdot2^{(2y^2)}ysin{(x^4)}^2ln{(2)})}}{z^2}}$

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

$\frac{{(72\cdot2^{(2y^2)}yln{(z^4)}sin{(x^4)}^2ln{(2)})}}{z}$

${-\frac{ {(12\cdot2^{(y^2)}sin{(x^4)})}}{z} - 6\cdot2^{(y^2)}yln{(z^4)}sin{(x^4)}ln{(2)}}$

Correct answer:

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

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[a^u]=a^uduln(a)\\&\text{Solve step-by-step:}\\&f_{z}=\frac{-{(12\cdot2^{(y^2)}sin{(x^4)})}}{z}\\&f_{zy}=\frac{-{(24\cdot2^{(y^2)}ysin{(x^4)}ln{(2)})}}{z}\end{align*}$

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Example Question #513 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yxzy}\\&\text{Where }f(x,y,z)=2\cdot3^zln{(x^2)}ln{(y^4)}\end{align*}$

Possible Answers:

$\frac{{(512\cdot3^{(4z)}ln{(x^2)}^3ln{(y^4)}^2ln{(3)})}}{{(xy^2)}}$

${\frac{{(16\cdot3^zln{(x^2)})}}{y} +\frac{ {(4\cdot3^zln{(y^4)})}}{x} + 2\cdot3^zln{(x^2)}ln{(y^4)}ln{(3)}}$

${\frac{-{(32768\cdot3^{(4z)}ln{(x^2)}ln{(3)}^2)}}{{(x^3y^5)}}}$

${\frac{-{(16\cdot3^zln{(3)})}}{{(xy^2)}}}$

Correct answer:

${\frac{-{(16\cdot3^zln{(3)})}}{{(xy^2)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[a^u]=a^uduln(a)\\&\text{Solve step-by-step:}\\&f_{y}=\frac{{(8\cdot3^zln{(x^2)})}}{y}\\&f_{yx}=\frac{{(16\cdot3^z)}}{{(xy)}}\\&f_{yxz}=\frac{{(16\cdot3^zln{(3)})}}{{(xy)}}\\&f_{yxzy}=\frac{-{(16\cdot3^zln{(3)})}}{{(xy^2)}}\end{align*}$

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Example Question #514 : Partial Derivatives

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

Possible Answers:

${\frac{{(54z^2)}}{x}}$

${\frac{{(27y^2z^3)}}{x} +\frac{ {(9y^3z^2)}}{x}}$

$\frac{{(6561y^9z^11)}}{x^4}$

${\frac{{(472392y^4z^10)}}{x^4}}$

Correct answer:

${\frac{{(54z^2)}}{x}}$

Explanation:

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Example Question #515 : Partial Derivatives

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

Possible Answers:

$2000z^4cos{(4y)}sin{(4x)}^3sin{(4y)}^2$

${16000z^2cos{(4y)}^2sin{(4x)}^3sin{(4y)}}$

${20zsin{(4x)}sin{(4y)} + 20z^2cos{(4y)}sin{(4x)}}$

{40cos{(4y)}sin{(4x)}}

Correct answer:

{40cos{(4y)}sin{(4x)}}

Explanation:

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Example Question #516 : Partial Derivatives

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

Possible Answers:

$\frac{{(9e^{(8z)}e^{(2y^2)})}}{x^2}$

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

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

${\frac{-{(9e^{(8z)}e^{(2y^2)})}}{x^3}}$

Correct answer:

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

Explanation:

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Example Question #517 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zxx}\\&\text{Where }f(x,y,z)=8cos{(3x)}cos{(z)}sin{(y)}\end{align*}$

Possible Answers:

{72cos{(3x)}sin{(y)}sin{(z)}}

${-4608cos{(3x)}sin{(3x)}^2cos{(z)}^2sin{(y)}^3sin{(z)}}$

${-13824cos{(3x)}^2sin{(3x)}sin{(y)}^3sin{(z)}^3}$

{- 8cos{(3x)}sin{(y)}sin{(z)} - 48sin{(3x)}cos{(z)}sin{(y)}}

Correct answer:

{72cos{(3x)}sin{(y)}sin{(z)}}

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f_{z}=-8cos{(3x)}sin{(y)}sin{(z)}\\&f_{zx}=24sin{(3x)}sin{(y)}sin{(z)}\\&f_{zxx}=72cos{(3x)}sin{(y)}sin{(z)}\end{align*}$

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Example Question #518 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yzzy}\\&\text{Where }f(x,y,z)=\frac{-{(8\cdot3^{(3z)}ln{(3y)})}}{x^2}\end{align*}$

Possible Answers:

${\frac{-{(995328\cdot3^{(12z)}ln{(3)}^5)}}{{(x^8y^5)}}}$

${-\frac{ {(16\cdot3^{(3z)})}}{{(x^2y)}} -\frac{ {(48\cdot3^{(3z)}ln{(3y)}ln{(3)})}}{x^2}}$

${\frac{{(36864\cdot3^{(12z)}ln{(3y)}^2ln{(3)}^2)}}{{(x^8y^2)}}}$

${\frac{{(72\cdot3^{(3z)}ln{(3)}^2)}}{{(x^2y^2)}}}$

Correct answer:

${\frac{{(72\cdot3^{(3z)}ln{(3)}^2)}}{{(x^2y^2)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[a^u]=a^uduln(a)\\&\text{Solve step-by-step:}\\&f_{y}=\frac{-{(8\cdot3^{(3z)})}}{{(x^2y)}}\\&f_{yz}=\frac{-{(24\cdot3^{(3z)}ln{(3)})}}{{(x^2y)}}\\&f_{yzz}=\frac{-{(72\cdot3^{(3z)}ln{(3)}^2)}}{{(x^2y)}}\\&f_{yzzy}=\frac{{(72\cdot3^{(3z)}ln{(3)}^2)}}{{(x^2y^2)}}\end{align*}$

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Example Question #511 : Partial Derivatives

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

Possible Answers:

${-32000x^2z^4ln{(y^2)}^3}$

${-40zln{(y^2)}}$

${- 40xz^2ln{(y^2)} - 20x^2zln{(y^2)}}$

${-8000x^4z^5ln{(y^2)}^3}$

Correct answer:

${-40zln{(y^2)}}$

Explanation:

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #511 : Partial Derivatives

Example Question #512 : Partial Derivatives

Example Question #512 : Partial Derivatives

Example Question #513 : Partial Derivatives

Example Question #514 : Partial Derivatives

Example Question #515 : Partial Derivatives

Example Question #516 : Partial Derivatives

Example Question #517 : Partial Derivatives

Example Question #518 : Partial Derivatives

Example Question #511 : Partial Derivatives

All Calculus 3 Resources

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