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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #3374 : Calculus 3

Find $f_{xxz}$ of the following function:

$f(x,y,z)=(4\arctan x)yz^2$

Possible Answers:

$f_{xxz}=\frac{16xyz}{(1+x^2)^2}$

$f_{xxz}=-\frac{16xyz}{(1+x^2)^2}$

$f_{xxz}=-\frac{16xyz}{(1+x^2)}$

$f_{xxz}=-\frac{16xyz^2}{(1+x^2)^2}$

Correct answer:

$f_{xxz}=-\frac{16xyz}{(1+x^2)^2}$

Explanation:

The way to solve this problem is by taking consecutive partial derivatives while everything else in the function behaves as a constant and is therefore not differentiated. For example, you can start by taking the partial derivative of the function with respect to x:

$f_{x}=\frac{4yz^2}{1+x^2}$

Then, you can take the partial derivative with respect to x again because x is expressed twice in the equation:

$f_{xx}=-\frac{8xyz^2}{(1+x^2)^2}$

Finally, you must take the partial derivative of the function with respect to z to get the final answer:

$f_{xxz}=-\frac{16xyz}{(1+x^2)^2}$

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Example Question #3375 : Calculus 3

Find $f_{xxy}$ of the following function:

$f(x,y,z)=(\frac{2xz}{x+1})\tan y$

Possible Answers:

$f_{xxy}=-\frac{4xz\sec^2 y}{(x+1)^2}$

$f_{xxy}=\frac{4z\sec^2 y}{(x+1)^3}$

$f_{xxy}=-\frac{4z\sec^2 y}{(x+1)^3}$

$f_{xxy}=-\frac{4\sec^2 y}{(x+1)^3}$

Correct answer:

$f_{xxy}=-\frac{4z\sec^2 y}{(x+1)^3}$

Explanation:

$f_{x}=\frac{2z\tan y}{(x+1)^2}$

Then, you can take the partial derivative with respect to x again because x is expressed twice in the equation:

$f_{xx}=-\frac{4z\tan y}{(x+1)^3}$

Finally, you must take the partial derivative of the function with respect to y to get the final answer:

$f_{xxy}=-\frac{4z\sec^2 y}{(x+1)^3}$

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Example Question #1011 : Partial Derivatives

Find $f_{yzz}$ of the following function:

$f(x,y,z)=8x^3y^4\ln z$

Possible Answers:

$f_{yzz}=-\frac{16x^3y^3}{z^2}$

$f_{yzz}=-\frac{32x^3y^2}{z^2}$

$f_{yzz}=-\frac{32x^3y^3}{z^2}$

$f_{yzz}=\frac{32x^3y^3}{z^2}$

Correct answer:

$f_{yzz}=-\frac{32x^3y^3}{z^2}$

Explanation:

$f_{y}=32x^3y^3\ln z$

Then, you can take the partial derivative with respect to z:

$f_{yz}=\frac{32x^3y^3}{z}$

Finally, you must take the partial derivative of the function with respect to z again to get the final answer because z is expressed twice in the question:

$f_{yzz}=-\frac{32x^3y^3}{z^2}$

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Example Question #1012 : Partial Derivatives

Find $f_{xyy}$ of the following function:

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

Possible Answers:

$f_{xyy}=\frac{18xe^{3y}z^2}{x^2+2}$

$f_{xyy}=\frac{6xe^{3y}z^2}{x^2+2}$

$f_{xyy}=-\frac{18xe^{3y}z^2}{x^2+2}$

$f_{xyy}=9e^{3y}z^2$

Correct answer:

$f_{xyy}=\frac{18xe^{3y}z^2}{x^2+2}$

Explanation:

$f_{x}=\frac{2xe^{3y}z^2}{x^2+2}$

Then, you can take the partial derivative with respect to y:

$f_{xy}=\frac{6xe^{3y}z^2}{x^2+2}$

Finally, you must take the partial derivative of the function with respect to z again to get the final answer because z is expressed twice in the question:

$f_{xyy}=\frac{18xe^{3y}z^2}{x^2+2}$

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Example Question #1013 : Partial Derivatives

Find $f_{yy}$ of the following function:

$f(x, y, z)=\frac{xy^2}{z}$

Possible Answers:

$-\frac{2x}{z}$

$\frac{2xy}{z}$

$\frac{2x}{z}$

Correct answer:

$\frac{2x}{z}$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

First, we find the partial derivative of the function with respect to y:

$f_y=\frac{2xy}{z}$

Finally, we take the derivative of the function above with respect to y:

$f_{yy}=\frac{2x}{z}$

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Example Question #1011 : Partial Derivatives

Find f_z of the function given:

$f(x, y, z)=e^{yz}\sin(z)+z^2+xz$

Possible Answers:

$e^{yz}(\sin(z)+\cos(z))+2z+x$

$ye^{yz}\sin(z)-e^{yz}\cos(z)+2z+x$

$ye^{yz}\sin(z)+e^{yz}\cos(z)+2z+x$

$ye^{yz}\sin(z)+e^{yz}\cos(z)+z(2+x)$

Correct answer:

$ye^{yz}\sin(z)+e^{yz}\cos(z)+2z+x$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivative of the function with respect to z is

$f_z=ye^{yz}\sin(z)+e^{yz}\cos(z)+2z+x$

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Example Question #1015 : Partial Derivatives

Find $f_{zz}$ of the following function:

$f(x, y, z)=z^3+xy\sin(z)$

Possible Answers:

$6z-xy\sin(z)$

$6z-\sin(z)$

$3y^2+xy\cos(z)$

$6z+xy\sin(z)$

Correct answer:

$6z-xy\sin(z)$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we find the partial derivative of the function with respect to z:

$f_z=3z^2+xy\cos(z)$

Finally, we find the derivative of the function above with respect to z:

$f_{zz}=6z-xy\sin(z)$

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Example Question #1016 : Partial Derivatives

Find $f_{xxyy}$ of the following function:

$f(x, y, z)=x^2y^2(z^2+\cos(e^z))$

Possible Answers:

$z^2+\cos(e^z)$

$4z^2+4\cos(e^z)$

$4y(z^2+\cos(e^z))$

Correct answer:

$4z^2+4\cos(e^z)$

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative of the function with respect to y:

$f_x=2xy^2(z^2+\cos(e^z))$

Next, we take the partial derivative of the above function with respect to x:

$f_{xx}=2y^2(z^2+\cos(e^z))$

Now, we take the partial derivative of the above function with respect to y:

$f_{xxy}=4y(z^2+\cos(e^z))$

Finally, we find the partial derivative of the function above with respect to y:

$f_{xxyy}=4z^2+4\cos(e^z)$

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Example Question #1017 : Partial Derivatives

Find $f_{xxy}$ of the function $x^2y+\sin(x)+x^2yz^3$

Possible Answers:

1+5z^3

2+2z^3

2+z^3

6+2z^2

Correct answer:

2+2z^3

Explanation:

To find $f_{xxy}$ of the function, we take three consecutive partial derivatives:

$\frac{\partial }{\partial x}(x^2y+\sin(x)+x^2yz^3)=2xy+\cos(x)+2xyz^3$

$\frac{\partial }{\partial x}(2xy+\cos(x)+2xyz^3)=2y-\sin(x)+2yz^3$

$\frac{\partial }{\partial y}(2y-\sin(x)+2yz^3)=2+2z^3$

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Example Question #1018 : Partial Derivatives

Find $f_{xxy}$ of the expression 3x^5y^7z^8+2x^2y^2

Possible Answers:

360x^2y^6z^8+7y

420x^3y^6z^8+8y

130x^5y^6z^6+7y

130x^3y^5z^8+8y

Correct answer:

420x^3y^6z^8+8y

Explanation:

To find $f_{xxy}$ of the function, we take three consecutive partial derivatives:

$\frac{\partial }{\partial x}(3x^5y^7z^8+2x^2y^2)=15x^4y^7z^8+4xy^2$

$\frac{\partial }{\partial x}(15x^4y^7z^8+4xy^2)=60x^3y^7z^8+4y^2$

$\frac{\partial }{\partial y}(60x^3y^7z^8+4y^2)=420x^3y^6z^8+8y$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #3374 : Calculus 3

Example Question #3375 : Calculus 3

Example Question #1011 : Partial Derivatives

Example Question #1012 : Partial Derivatives

Example Question #1013 : Partial Derivatives

Example Question #1011 : Partial Derivatives

Example Question #1015 : Partial Derivatives

Example Question #1016 : Partial Derivatives

Example Question #1017 : Partial Derivatives

Example Question #1018 : Partial Derivatives

All Calculus 3 Resources

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