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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #611 : Partial Derivatives

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

Possible Answers:

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

${2x^{2}y^{2} - 6zcos(y^{3})sin(z^{3})e^{(x)} - 9z^{4}cos(y^{3})cos(z^{3})e^{(x)}}$

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

${-(2x^{2}y^{2}z - 3z^{2}cos(y^{3})sin(z^{3})e^{(x)})\cdot(6zcos(y^{3})sin(z^{3})e^{(x)} - 2x^{2}y^{2} + 9z^{4}cos(y^{3})cos(z^{3})e^{(x)})}$

Correct answer:

${2x^{2}y^{2} - 6zcos(y^{3})sin(z^{3})e^{(x)} - 9z^{4}cos(y^{3})cos(z^{3})e^{(x)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=cos(y^{3})cos(z^{3})e^{(x)} + x^{2}y^{2}z^{2}\\&f_{z}=2x^{2}y^{2}z - 3z^{2}cos(y^{3})sin(z^{3})e^{(x)}\\&f_{zz}=2x^{2}y^{2} - 6zcos(y^{3})sin(z^{3})e^{(x)} - 9z^{4}cos(y^{3})cos(z^{3})e^{(x)}\end{align*}$

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Example Question #612 : Partial Derivatives

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

Possible Answers:

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

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

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

${-(2^{(3z)}ln(4x)e^{(3y)} - 2yz^{2}e^{(x)})\cdot(2z^{2}e^{(x)} - 3\cdot2^{(3z)}ln(4x)e^{(3y)})}$

Correct answer:

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

Explanation:

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Example Question #613 : Partial Derivatives

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

Possible Answers:

${2zcos(y) -\frac{ (2e^{(3y)})}{(9x^{2}z)}+\frac{ (4e^{(3y)}ln(z))}{(9x^{3})}}$

${\frac{(4e^{(3y)}ln(z)\cdot(2zcos(y) -\frac{ (2e^{(3y)})}{(9x^{2}z)}))}{(9x^{3})}}$

${\frac{(4e^{(3y)})}{(9x^{3}z)}}$

${\frac{(16e^{(6y)}ln(z))}{(81x^{6}z)}}$

Correct answer:

${\frac{(4e^{(3y)})}{(9x^{3}z)}}$

Explanation:

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Example Question #614 : Partial Derivatives

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

Possible Answers:

${-(\frac{4}{(xy^{2})}-\frac{ (2cos(3y)ln(4z))}{(3x)})\cdot(\frac{(8ln(x))}{y^{3}}- 2ln(4z)sin(3y)ln(x))}$

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

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

${-(\frac{4}{(xy^{2})}-\frac{ (2cos(3y)ln(4z))}{(3x)})\cdot(\frac{8}{(xy^{3})}-\frac{ (2ln(4z)sin(3y))}{x})}$

Correct answer:

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

Explanation:

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

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Example Question #615 : Partial Derivatives

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

Possible Answers:

${-\frac{(cos(3y)ln(4z)sin(x)\cdot(\frac{(6sin(y))}{z}+\frac{ (cos(3y)cos(x))}{(4z)}))}{4}}$

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

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

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

Correct answer:

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

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=3ln(z^{2})sin(y) +\frac{ (cos(3y)ln(4z)cos(x))}{4}\\&f_{z}=\frac{(6sin(y))}{z}+\frac{ (cos(3y)cos(x))}{(4z)}\\&f_{zx}=-\frac{(cos(3y)sin(x))}{(4z)}\end{align*}$

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Example Question #616 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zy}\\&\text{Where }f(x,y,z)=4\cdot3^{(y^{2})}z^{2}ln(x) + 5ln(z^{2})sin(x^{3})e^{(y)}\end{align*}$

Possible Answers:

${\frac{(10sin(x^{3})e^{(y)})}{z}+ 5ln(z^{2})sin(x^{3})e^{(y)} + 8\cdot3^{(y^{2})}zln(x) + 8\cdot3^{(y^{2})}yz^{2}ln(3)ln(x)}$

${(\frac{(10sin(x^{3})e^{(y)})}{z}+ 16\cdot3^{(y^{2})}yzln(3)ln(x))\cdot(\frac{(10sin(x^{3})e^{(y)})}{z}+ 8\cdot3^{(y^{2})}zln(x))}$

${\frac{(10sin(x^{3})e^{(y)})}{z}+ 16\cdot3^{(y^{2})}yzln(3)ln(x)}$

${(5ln(z^{2})sin(x^{3})e^{(y)} + 8\cdot3^{(y^{2})}yz^{2}ln(3)ln(x))\cdot(\frac{(10sin(x^{3})e^{(y)})}{z}+ 8\cdot3^{(y^{2})}zln(x))}$

Correct answer:

${\frac{(10sin(x^{3})e^{(y)})}{z}+ 16\cdot3^{(y^{2})}yzln(3)ln(x)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[a^u]=a^uduln(a)\\&d[ln(u)]=\frac{du}{u}\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=4\cdot3^{(y^{2})}z^{2}ln(x) + 5ln(z^{2})sin(x^{3})e^{(y)}\\&f_{z}=\frac{(10sin(x^{3})e^{(y)})}{z}+ 8\cdot3^{(y^{2})}zln(x)\\&f_{zy}=\frac{(10sin(x^{3})e^{(y)})}{z}+ 16\cdot3^{(y^{2})}yzln(3)ln(x)\end{align*}$

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Example Question #617 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{zz}\\&\text{Where }f(x,y,z)=5x^{2}sin(z^{2})ln(y) -\frac{ (3\cdot4^{(4x)}cos(y^{2})ln(4z))}{2}\end{align*}$

Possible Answers:

${10x^{2}cos(z^{2})ln(y) +\frac{ (3\cdot4^{(4x)}cos(y^{2}))}{(2z^{2})}- 20x^{2}z^{2}sin(z^{2})ln(y)}$

${-(\frac{(3\cdot4^{(4x)}cos(y^{2}))}{(2z)}- 10x^{2}zcos(z^{2})ln(y))\cdot(10x^{2}cos(z^{2})ln(y) +\frac{ (3\cdot4^{(4x)}cos(y^{2}))}{(2z^{2})}- 20x^{2}z^{2}sin(z^{2})ln(y))}$

${(\frac{(3\cdot4^{(4x)}cos(y^{2}))}{(2z)}- 10x^{2}zcos(z^{2})ln(y))^{2}}$

${20x^{2}zcos(z^{2})ln(y) -\frac{ (3\cdot4^{(4x)}cos(y^{2}))}{z}}$

Correct answer:

${10x^{2}cos(z^{2})ln(y) +\frac{ (3\cdot4^{(4x)}cos(y^{2}))}{(2z^{2})}- 20x^{2}z^{2}sin(z^{2})ln(y)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&d[sin(u)]=cos(u)du\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=5x^{2}sin(z^{2})ln(y) -\frac{ (3\cdot4^{(4x)}cos(y^{2})ln(4z))}{2}\\&f_{z}=10x^{2}zcos(z^{2})ln(y) -\frac{ (3\cdot4^{(4x)}cos(y^{2}))}{(2z)}\\&f_{zz}=10x^{2}cos(z^{2})ln(y) +\frac{ (3\cdot4^{(4x)}cos(y^{2}))}{(2z^{2})}- 20x^{2}z^{2}sin(z^{2})ln(y)\end{align*}$

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Example Question #618 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yy}\\&\text{Where }f(x,y,z)=4cos(y^{2})sin(z^{3})e^{(x)} -\frac{ (2cos(3y + y^{4} - 1)sin(z))}{x}\end{align*}$

Possible Answers:

${\frac{(4sin(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3))}{x}- 16ysin(y^{2})sin(z^{3})e^{(x)}}$

${-(\frac{(2sin(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3))}{x}- 8ysin(y^{2})sin(z^{3})e^{(x)})\cdot(8sin(y^{2})sin(z^{3})e^{(x)} -\frac{ (24y^{2}sin(3y + y^{4} - 1)sin(z))}{x}-\frac{ (2cos(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3)^{2})}{x}+ 16y^{2}cos(y^{2})sin(z^{3})e^{(x)})}$

${\frac{(24y^{2}sin(3y + y^{4} - 1)sin(z))}{x}- 8sin(y^{2})sin(z^{3})e^{(x)} +\frac{ (2cos(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3)^{2})}{x}- 16y^{2}cos(y^{2})sin(z^{3})e^{(x)}}$

${(\frac{(2sin(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3))}{x}- 8ysin(y^{2})sin(z^{3})e^{(x)})^{2}}$

Correct answer:

${\frac{(24y^{2}sin(3y + y^{4} - 1)sin(z))}{x}- 8sin(y^{2})sin(z^{3})e^{(x)} +\frac{ (2cos(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3)^{2})}{x}- 16y^{2}cos(y^{2})sin(z^{3})e^{(x)}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[cos(u)]=-sin(u)du\\&d[sin(u)]=cos(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=4cos(y^{2})sin(z^{3})e^{(x)} -\frac{ (2cos(3y + y^{4} - 1)sin(z))}{x}\\&f_{y}=\frac{(2sin(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3))}{x}- 8ysin(y^{2})sin(z^{3})e^{(x)}\\&f_{yy}=\frac{(24y^{2}sin(3y + y^{4} - 1)sin(z))}{x}- 8sin(y^{2})sin(z^{3})e^{(x)} +\frac{ (2cos(3y + y^{4} - 1)sin(z)\cdot(4y^{3} + 3)^{2})}{x}- 16y^{2}cos(y^{2})sin(z^{3})e^{(x)}\end{align*}$

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Example Question #619 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yx}\\&\text{Where }f(x,y,z)=\frac{(3^{(x^{2})}\cdot3^ye^{(z)})}{3}+ 3xln(z^{2})sin(y^{3})\end{align*}$

Possible Answers:

${(9xy^{2}cos(y^{3})ln(z^{2}) +\frac{ (3^{(x^{2})}\cdot3^ye^{(z)}ln(3))}{3})\cdot(3ln(z^{2})sin(y^{3}) +\frac{ (2\cdot3^{(x^{2})}\cdot3^yxe^{(z)}ln(3))}{3})}$

${3ln(z^{2})sin(y^{3}) + 9xy^{2}cos(y^{3})ln(z^{2}) +\frac{ (3^{(x^{2})}\cdot3^ye^{(z)}ln(3))}{3}+\frac{ (2\cdot3^{(x^{2})}\cdot3^yxe^{(z)}ln(3))}{3}}$

${(9xy^{2}cos(y^{3})ln(z^{2}) +\frac{ (3^{(x^{2})}\cdot3^ye^{(z)}ln(3))}{3})\cdot(9y^{2}cos(y^{3})ln(z^{2}) +\frac{ (2\cdot3^{(x^{2})}\cdot3^yxe^{(z)}ln(3)^{2})}{3})}$

${9y^{2}cos(y^{3})ln(z^{2}) +\frac{ (2\cdot3^{(x^{2})}\cdot3^yxe^{(z)}ln(3)^{2})}{3}}$

Correct answer:

${9y^{2}cos(y^{3})ln(z^{2}) +\frac{ (2\cdot3^{(x^{2})}\cdot3^yxe^{(z)}ln(3)^{2})}{3}}$

Explanation:

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

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Example Question #620 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xx}\\&\text{Where }f(x,y,z)=4x^{3}y^{3}z +\frac{ (2\cdot4^{(3z)}ln(x^{2})ln(y))}{3}\end{align*}$

Possible Answers:

${24xy^{3}z -\frac{ (4\cdot4^{(3z)}ln(y))}{(3x^{2})}}$

${\frac{(8\cdot4^{(3z)}ln(y))}{(3x)}+ 24x^{2}y^{3}z}$

${(\frac{(4\cdot4^{(3z)}ln(y))}{(3x)}+ 12x^{2}y^{3}z)^{2}}$

${(\frac{(4\cdot4^{(3z)}ln(y))}{(3x)}+ 12x^{2}y^{3}z)\cdot(24xy^{3}z -\frac{ (4\cdot4^{(3z)}ln(y))}{(3x^{2})})}$

Correct answer:

${24xy^{3}z -\frac{ (4\cdot4^{(3z)}ln(y))}{(3x^{2})}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=4x^{3}y^{3}z +\frac{ (2\cdot4^{(3z)}ln(x^{2})ln(y))}{3}\\&f_{x}=\frac{(4\cdot4^{(3z)}ln(y))}{(3x)}+ 12x^{2}y^{3}z\\&f_{xx}=24xy^{3}z -\frac{ (4\cdot4^{(3z)}ln(y))}{(3x^{2})}\end{align*}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #611 : Partial Derivatives

Example Question #612 : Partial Derivatives

Example Question #613 : Partial Derivatives

Example Question #614 : Partial Derivatives

Example Question #615 : Partial Derivatives

Example Question #616 : Partial Derivatives

Example Question #617 : Partial Derivatives

Example Question #618 : Partial Derivatives

Example Question #619 : Partial Derivatives

Example Question #620 : Partial Derivatives

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