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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #571 : Partial Derivatives

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

Possible Answers:

${-6cos{(y^2)}e^{(3x)}e^{(z)}}$

${-24cos{(y^2)}^3e^{(9x)}e^{(3z)}}$

${-10cos{(y^2)}e^{(3x)}e^{(z)}}$

Correct answer:

${-6cos{(y^2)}e^{(3x)}e^{(z)}}$

Explanation:

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

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Example Question #572 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zyzx}\\&\text{Where }f(x,y,z)=\frac{-{(2\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)})}}{9}\end{align*}$

Possible Answers:

${\frac{{(32\cdot2^{(12z)}\cdot3^{(4x^2)}xe^{(4y)}ln{(2)}^6ln{(3)})}}{9}}$

${-\frac{ {(2\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)})}}{9} -\frac{ {(4\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)}ln{(2)})}}{3} -\frac{ {(4\cdot2^{(3z)}\cdot3^{(x^2)}xe^{(y)}ln{(3)})}}{9}}$

${-4\cdot2^{(3z)}\cdot3^{(x^2)}xe^{(y)}ln{(2)}^2ln{(3)}}$

${\frac{{(32\cdot2^{(12z)}\cdot3^{(4x^2)}xe^{(4y)}ln{(2)}^2ln{(3)})}}{729}}$

Correct answer:

${-4\cdot2^{(3z)}\cdot3^{(x^2)}xe^{(y)}ln{(2)}^2ln{(3)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[a^u]=a^uduln(a)\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{-{(2\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)})}}{9}\\&f_{z}=\frac{-{(2\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)}ln{(2)})}}{3}\\&f_{zy}=\frac{-{(2\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)}ln{(2)})}}{3}\\&f_{zyz}=-2\cdot2^{(3z)}\cdot3^{(x^2)}e^{(y)}ln{(2)}^2\\&f_{zyzx}=-4\cdot2^{(3z)}\cdot3^{(x^2)}xe^{(y)}ln{(2)}^2ln{(3)}\end{align*}$

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Example Question #573 : Partial Derivatives

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

Possible Answers:

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

${-64cos{(4y)}ln{(z^2)}^2sin{(4x)}^2sin{(4y)}}$

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

${16ln{(z^2)}sin{(4x)}sin{(4y)}}$

Correct answer:

${16ln{(z^2)}sin{(4x)}sin{(4y)}}$

Explanation:

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Example Question #574 : Partial Derivatives

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

Possible Answers:

${\frac{{(8z^2e^{(4y)})}}{x^3}}$

${\frac{-{(48z^4e^{(8y)})}}{x^7}}$

${\frac{-{(12z^2e^{(4y)})}}{x^4}}$

${\frac{{(16z^4e^{(8y)})}}{x^6}}$

Correct answer:

${\frac{-{(12z^2e^{(4y)})}}{x^4}}$

Explanation:

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Example Question #575 : Partial Derivatives

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

Possible Answers:

${\frac{{(3\cdot4^{(6y)}xcos{(3z)}sin{(3z)}e^{(2x^2)})}}{2}}$

${-\frac{ {(3\cdot4^{(3y)}cos{(3z)}e^{(x^2)})}}{2} - 4^{(3y)}xsin{(3z)}e^{(x^2)}}$

${-3\cdot4^{(3y)}xcos{(3z)}e^{(x^2)}}$

${\frac{{(9\cdot4^{(6y)}xcos{(3z)}^2e^{(2x^2)})}}{2}}$

Correct answer:

${-3\cdot4^{(3y)}xcos{(3z)}e^{(x^2)}}$

Explanation:

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

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Example Question #576 : Partial Derivatives

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

Possible Answers:

${\frac{-{(4cos{(4y)}cos{(3z)}sin{(4y)}^2sin{(3z)}^2)}}{{(9x^4)}}}$

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

${\frac{-{(4cos{(4y)}cos{(3z)}^2sin{(4y)}^2sin{(3z)})}}{{(3x^6)}}}$

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

Correct answer:

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

Explanation:

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

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Example Question #577 : Partial Derivatives

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

Possible Answers:

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

${18z^3cos{(3x)}^2cos{(4y)}^2}$

{6zcos{(3x)}cos{(4y)}}

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

Correct answer:

{6zcos{(3x)}cos{(4y)}}

Explanation:

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Example Question #578 : Partial Derivatives

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

Possible Answers:

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

${\frac{-{(z^4cos{(3y)}^3ln{(3x)}^2)}}{{(16x)}}}$

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

${\frac{-cos{(3y)}}{{(2x)}}}$

Correct answer:

${\frac{-cos{(3y)}}{{(2x)}}}$

Explanation:

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

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Example Question #579 : Partial Derivatives

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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Example Question #580 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yx}\\&\text{Where }f(x,y,z)=\frac{-{(2\cdot4^xln{(y)}sin{(z)})}}{7}\end{align*}$

Possible Answers:

${-\frac{ {(2\cdot4^xsin{(z)})}}{{(7y)}} -\frac{ {(2\cdot4^xln{(4)}ln{(y)}sin{(z)})}}{7}}$

${\frac{{(4\cdot4^{(2x)}ln{(4)}ln{(y)}sin{(z)}^2)}}{{(49y)}}}$

${\frac{{(4\cdot4^{(2x)}ln{(4)}sin{(z)}^2)}}{{(49y^2)}}}$

${\frac{-{(2\cdot4^xln{(4)}sin{(z)})}}{{(7y)}}}$

Correct answer:

${\frac{-{(2\cdot4^xln{(4)}sin{(z)})}}{{(7y)}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[a^u]=a^uduln(a)\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{-{(2\cdot4^xln{(y)}sin{(z)})}}{7}\\&f_{y}=\frac{-{(2\cdot4^xsin{(z)})}}{{(7y)}}\\&f_{yx}=\frac{-{(2\cdot4^xln{(4)}sin{(z)})}}{{(7y)}}\end{align*}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #571 : Partial Derivatives

Example Question #572 : Partial Derivatives

Example Question #573 : Partial Derivatives

Example Question #574 : Partial Derivatives

Example Question #575 : Partial Derivatives

Example Question #576 : Partial Derivatives

Example Question #577 : Partial Derivatives

Example Question #578 : Partial Derivatives

Example Question #579 : Partial Derivatives

Example Question #580 : Partial Derivatives

All Calculus 3 Resources

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