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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #681 : Partial Derivatives

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #682 : Partial Derivatives

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

Possible Answers:

${3x^{3}z^{2}sin(y) - 20e^{(4y)}e^{(z^{2})}ln(x) - 6x^{3}zcos(y) - 10ze^{(4y)}e^{(z^{2})}ln(x)}$

${6x^{3}zsin(y) - 40ze^{(4y)}e^{(z^{2})}ln(x)}$

${(20e^{(4y)}e^{(z^{2})}ln(x) - 3x^{3}z^{2}sin(y))\cdot (6x^{3}zcos(y) + 10ze^{(4y)}e^{(z^{2})}ln(x))}$

${-(20e^{(4y)}e^{(z^{2})}ln(x) - 3x^{3}z^{2}sin(y))\cdot (6x^{3}zsin(y) - 40ze^{(4y)}e^{(z^{2})}ln(x))}$

Correct answer:

${6x^{3}zsin(y) - 40ze^{(4y)}e^{(z^{2})}ln(x)}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[e^u]=e^udu\\&d[cos(u)]=-sin(u)du\\&\text{Solve step-by-step:}\\&f(x,y,z)=- 5e^{(4y)}e^{(z^{2})}ln(x) - 3x^{3}z^{2}cos(y)\\&f_{y}=3x^{3}z^{2}sin(y) - 20e^{(4y)}e^{(z^{2})}ln(x)\\&f_{yz}=6x^{3}zsin(y) - 40ze^{(4y)}e^{(z^{2})}ln(x)\end{align*}$

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Example Question #683 : Partial Derivatives

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

Possible Answers:

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

${\frac{(4e^{(z)})}{y}- 8\cdot 4^{(y^{2})}e^{(4x)}cos(z) - 4\cdot 4^{(y^{2})}ye^{(4x)}ln(4)cos(z)}$

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

${-8\cdot 4^{(y^{2})}e^{(4x)}cos(z)\cdot (\frac{(4e^{(z)})}{y}- 4\cdot 4^{(y^{2})}ye^{(4x)}ln(4)cos(z))}$

Correct answer:

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

Explanation:

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

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Example Question #684 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xy}\\&\text{Where }f(x,y,z)=\frac{(2sin(x^{2})e^{(z^{2})}tan(11y + y^{3}))}{7}-\frac{ (z^{4}ln(x^{2}))}{y^{3}}\end{align*}$

Possible Answers:

${\frac{(3z^{4}ln(x^{2}))}{y^{4}}-\frac{ (2z^{4})}{(xy^{3})}+\frac{ (2sin(x^{2})e^{(z^{2})}\cdot (tan(11y + y^{3})^{2} + 1)\cdot (3y^{2} + 11))}{7}+\frac{ (4xcos(x^{2})e^{(z^{2})}tan(11y + y^{3}))}{7}}$

${\frac{(6z^{4})}{(xy^{4})}+\frac{ (4xcos(x^{2})e^{(z^{2})}\cdot (tan(11y + y^{3})^{2} + 1)\cdot (3y^{2} + 11))}{7}}$

${-(\frac{(2z^{4})}{(xy^{3})}-\frac{ (4xcos(x^{2})e^{(z^{2})}tan(11y + y^{3}))}{7})\cdot (\frac{(3z^{4}ln(x^{2}))}{y^{4}}+\frac{ (2sin(x^{2})e^{(z^{2})}\cdot (tan(11y + y^{3})^{2} + 1)\cdot (3y^{2} + 11))}{7})}$

${-(\frac{(2z^{4})}{(xy^{3})}-\frac{ (4xcos(x^{2})e^{(z^{2})}tan(11y + y^{3}))}{7})\cdot (\frac{(6z^{4})}{(xy^{4})}+\frac{ (4xcos(x^{2})e^{(z^{2})}\cdot (tan(11y + y^{3})^{2} + 1)\cdot (3y^{2} + 11))}{7})}$

Correct answer:

${\frac{(6z^{4})}{(xy^{4})}+\frac{ (4xcos(x^{2})e^{(z^{2})}\cdot (tan(11y + y^{3})^{2} + 1)\cdot (3y^{2} + 11))}{7}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[sin(u)]=cos(u)du\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&d[ln(u)]=\frac{du}{u}\\&\text{Solve step-by-step:}\\&f(x,y,z)=\frac{(2sin(x^{2})e^{(z^{2})}tan(11y + y^{3}))}{7}-\frac{ (z^{4}ln(x^{2}))}{y^{3}}\\&f_{x}=\frac{(4xcos(x^{2})e^{(z^{2})}tan(11y + y^{3}))}{7}-\frac{ (2z^{4})}{(xy^{3})}\\&f_{xy}=\frac{(6z^{4})}{(xy^{4})}+\frac{ (4xcos(x^{2})e^{(z^{2})}\cdot (tan(11y + y^{3})^{2} + 1)\cdot (3y^{2} + 11))}{7}\end{align*}$

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Example Question #685 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{xy}\\&\text{Where }f(x,y,z)=5e^{(y^{4})}e^{(z^{3})} +\frac{ (4e^{(3z)}tan(x^{2} - 12x))}{y^{2}}\end{align*}$

Possible Answers:

${-\frac{(8e^{(3z)}\cdot (2x - 12)\cdot (tan(x^{2} - 12x)^{2} + 1))}{y^{3}}}$

${20y^{3}e^{(y^{4})}e^{(z^{3})} -\frac{ (8e^{(3z)}tan(x^{2} - 12x))}{y^{3}}+\frac{ (4e^{(3z)}\cdot (2x - 12)\cdot (tan(x^{2} - 12x)^{2} + 1))}{y^{2}}}$

${-\frac{(32e^{(6z)}\cdot (2x - 12)^{2}\cdot (tan(x^{2} - 12x)^{2} + 1)^{2})}{y^{5}}}$

${\frac{(4e^{(3z)}\cdot (20y^{3}e^{(y^{4})}e^{(z^{3})} -\frac{ (8e^{(3z)}tan(x^{2} - 12x))}{y^{3}})\cdot (2x - 12)\cdot (tan(x^{2} - 12x)^{2} + 1))}{y^{2}}}$

Correct answer:

${-\frac{(8e^{(3z)}\cdot (2x - 12)\cdot (tan(x^{2} - 12x)^{2} + 1))}{y^{3}}}$

Explanation:

$\begin{align*}&\text{When taking partial derivatives, order does not matter; you can}\\&\text{simply treat any variable that you're not deriving the equation for as constant.}\\&\text{Utilizing derivative rules:}\\&d[uv]=udv+vdu\\&(f\circ g )'=(f'\circ g )\cdot g'\\&d[tan(u)]=\frac{du}{cos^2(u)}=(tan^2(u)+1)du\\&d[e^u]=e^udu\\&\text{Solve step-by-step:}\\&f(x,y,z)=5e^{(y^{4})}e^{(z^{3})} +\frac{ (4e^{(3z)}tan(x^{2} - 12x))}{y^{2}}\\&f_{x}=\frac{(4e^{(3z)}\cdot (2x - 12)\cdot (tan(x^{2} - 12x)^{2} + 1))}{y^{2}}\\&f_{xy}=-\frac{(8e^{(3z)}\cdot (2x - 12)\cdot (tan(x^{2} - 12x)^{2} + 1))}{y^{3}}\end{align*}$

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Example Question #686 : Partial Derivatives

$\begin{align*}&\text{Perform the partial derivation, }f_{yy}\\&\text{Where }f(x,y,z)=\frac{(9\cdot 4^xy^{2}z^{2})}{2}- 3x^{4}ysin(z)\end{align*}$

Possible Answers:

${18\cdot 4^xyz^{2} - 6x^{4}sin(z)}$

${9\cdot 4^xz^{2}}$

${(3x^{4}sin(z) - 9\cdot 4^xyz^{2})^{2}}$

${-9\cdot 4^xz^{2}\cdot (3x^{4}sin(z) - 9\cdot 4^xyz^{2})}$

Correct answer:

${9\cdot 4^xz^{2}}$

Explanation:

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Example Question #687 : Partial Derivatives

Find the partial derivative

$\frac{\partial f}{\partial x}$

of the function

$f(x,y)=e^{2x+3y}$

Possible Answers:

$\frac{\partial f}{\partial x}=5e^{2x+3y}$

$\frac{\partial f}{\partial x}=2e^{2x+3y}$

$\frac{\partial f}{\partial x}=3e^{2x+3y}$

$\frac{\partial f}{\partial x}=6e^{2x+3y}$

Correct answer:

$\frac{\partial f}{\partial x}=2e^{2x+3y}$

Explanation:

To find the partial derivative with respect to , or

$\frac{\partial f }{\partial x}$

we take the derivative with respect to and treat the other variables as constants.

$\frac{\partial f }{\partial x}=\frac{\partial }{\partial x}(e^{2x+3y})=2e^{2x+3y}$

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Example Question #688 : Partial Derivatives

Find the partial derivative

$\frac{\partial f}{\partial y}$

of the function

f(x,y)=xcos(y)

Possible Answers:

$\frac{\partial f}{\partial y}=-xsin(y)$

$\frac{\partial f}{\partial y}=xsin(y)$

$\frac{\partial f}{\partial y}=\frac{1}{2}x^2cos(y)$

$\frac{\partial f}{\partial y}=cos(y)$

Correct answer:

$\frac{\partial f}{\partial y}=-xsin(y)$

Explanation:

To find the partial derivative with respect to , or

$\frac{\partial f }{\partial y}$

we take the derivative with respect to and treat the other variables as constants.

$\frac{\partial f }{\partial y}=\frac{\partial }{\partial y}[xcos(y)]=-xsin(y)$

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Example Question #689 : Partial Derivatives

Find the partial derivative with respect to , $\frac{\partial F}{\partial x}$ , of the function

F(x,y)=cos( x) + sin (y) +2xy

Possible Answers:

$\frac{\partial F}{\partial x}=cos(y)+2x$

$\frac{\partial F}{\partial x}=cos(x)+cos(y)+2$

$\frac{\partial F}{\partial x}=-sin(x)+2y$

$\frac{\partial F}{\partial x}=-sin(x)-cos(y)$

Correct answer:

$\frac{\partial F}{\partial x}=-sin(x)+2y$

Explanation:

To find The partial derivative with respect to

$\frac{\partial F}{\partial x}$

you derive the function with respect to and treat the other variables as constants.

$\frac{\partial F}{\partial x}=\frac{\partial }{\partial x}[cos( x) + sin (y) +2xy]=-sin(x)+2y$

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Example Question #690 : Partial Derivatives

Find the partial derivative with respect to , $\frac{\partial F}{\partial y}$ , of the function

F(x,y)=tan (x) - cot (y) + e^y

Possible Answers:

$\frac{\partial F}{\partial y}=sec^2(x)$

$\frac{\partial F}{\partial y}=e^y$

$\frac{\partial F}{\partial y}=csc^2(y)+e^y$

$\frac{\partial F}{\partial y}=sex^2(x)+csc^2(y)+e^y$

Correct answer:

$\frac{\partial F}{\partial y}=csc^2(y)+e^y$

Explanation:

To find The partial derivative with respect to

$\frac{\partial F}{\partial y}$

you derive the function with respect to and treat the other variables as constants.

$\frac{\partial F}{\partial y}=\frac{\partial }{\partial y}[tan (x) - cot (y) + e^y]=csc^2(y)+e^y$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #681 : Partial Derivatives

Example Question #682 : Partial Derivatives

Example Question #683 : Partial Derivatives

Example Question #684 : Partial Derivatives

Example Question #685 : Partial Derivatives

Example Question #686 : Partial Derivatives

Example Question #687 : Partial Derivatives

Example Question #688 : Partial Derivatives

Example Question #689 : Partial Derivatives

Example Question #690 : Partial Derivatives

All Calculus 3 Resources

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