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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #3399 : Calculus 3

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

$f(x, y, z)=x+x^2y^2+xy^3+x\cos(z)$

Possible Answers:

2x^2+3x^2y

4x+6y

2x^2+6xy

Correct answer:

4x+6y

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=2x^2+3xy^2

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

$f_{yy}=2x^2+6xy$

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

$f_{yyx}=4x+6y$

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Example Question #3400 : Calculus 3

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

Possible Answers:

6y^2+3z

6y^2+6z^2

6y^2+4z

y^2+4z

Correct answer:

6y^2+4z

Explanation:

To find $f_{xxy}$ we take three consecutive partial derivatives

$\frac{\partial }{\partial x}(x^2y^3+2x^2yz)=2xy^3+4xyz$

$\frac{\partial }{\partial x}(2xy^3+4xyz)=2y^3+4yz$

$\frac{\partial }{\partial y}(2y^3+4yz)=6y^2+4z$

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Example Question #3401 : Calculus 3

Find $f_{xxy}$ of the function 3x^3y+x^2y^2

Possible Answers:

11x+4y

20x+4y^2

18x+4y

18x+5y

Correct answer:

18x+4y

Explanation:

To find $f_{xxy}$ we take three consecutive partial derivatives

$\frac{\partial }{\partial x}(3x^3y+x^2y^2)=9x^2y+2xy^2$

$\frac{\partial }{\partial x}(9x^2y+2xy^2)=18xy+2y^2$

$\frac{\partial }{\partial y}(18xy+2y^2)=18x+4y$

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Example Question #3402 : Calculus 3

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

$f(x,y,z)=10z\sec(xy)$

Possible Answers:

$10y\sec(xy)\tan(xy)$

$10\sec(xy)$

$-10x\sec(xy)\tan(xy)$

$10x\sec(xy)\tan(xy)$

Correct answer:

$10x\sec(xy)\tan(xy)$

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:

$f_z=10\sec(xy)$

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

$f_{zy}=10x\sec(xy)\tan(xy)$

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Example Question #3403 : Calculus 3

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

$f(x, y, z)=\ln(x^2y^2)+xyz$

Possible Answers:

$-\frac{2y}{x^2}$

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

$\frac{2}{x}+yz$

Correct answer:

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 x:

$f_x=\frac{2}{x}+yz$

Next, we find the partial derivative of this function with respect to x:

$f_{xx}=-\frac{2}{x^2}$

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

$f_{xxy}=0$

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Example Question #3404 : Calculus 3

Find $(f_{xy})^2$ of the following function:

Possible Answers:

$4x^2z^2(\cos^2(y)-y^2(\sin^2(y)))$

$4x^2z^2(\cos^2(y)+y^2(\sin^2(y)))$

$2xz(\cos(y)-y\sin(y))$

$4x^2z^2(\cos(y)-y\sin(y))^2$

Correct answer:

$4x^2z^2(\cos(y)-y\sin(y))^2$

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 x:

$f_x=2xy\cos(y)z$

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

$f_{xy}=2xz(\cos(y)-y\sin(y))$

Finally, we square this function:

$(f_{xy})^2=4x^2z^2(\cos(y)-y\sin(y))^2$

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Example Question #3405 : Calculus 3

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

f(x,y)=x^5ye^y

Possible Answers:

x^5e^y(1+y)

x^5e^y(2+y)

x^5(2e^y+y)

2x^5e^y

Correct answer:

x^5e^y(2+y)

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=x^5(e^y+ye^y)

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

$f_{yy}=x^5(e^y+e^y+ye^y)=x^5e^y(2+y)$

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Example Question #3406 : Calculus 3

Find $f_{xy}$ for the following function:

$f(x,y)=xy\cos(x^2)$

Possible Answers:

$\cos(x^2)-2x\sin(x^2)$

$y\cos(x^2)-2x^2y\sin(x^2)$

$\cos(x^2)+2x^2\sin(x^2)$

$\cos(x^2)-2x^2\sin(x^2)$

Correct answer:

$\cos(x^2)-2x^2\sin(x^2)$

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 x:

$f_x=y\cos(x^2)-2x^2y\sin(x^2)$

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

$f_{xy}=\cos(x^2)-2x^2\sin(x^2)$

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Example Question #3407 : Calculus 3

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

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

Possible Answers:

$-2y\sin(y^2)$

$z\cos(y^2)$

$\cos(y^2)$

Correct answer:

$-2y\sin(y^2)$

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 x:

$f_x=z\cos(y^2)$

Next, we find the partial derivative of this function with respect to z:

$f_{xz}=\cos(y^2)$

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

$f_{xzy}=-2y\sin(y^2)$

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Example Question #3408 : Calculus 3

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

$f(x,y,z)=\cos(xyz)$

Possible Answers:

$x^2z^2\cos(xyz)$

$-x^2z^2\cos(xyz)$

$-xz\sin(xyz)$

$-x^2z^2\sin(xyz)$

Correct answer:

$-x^2z^2\cos(xyz)$

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:

$f_y=-xz\sin(xyz)$

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

$f_{yy}=-x^2z^2\cos(xyz)$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #3399 : Calculus 3

Example Question #3400 : Calculus 3

Example Question #3401 : Calculus 3

Example Question #3402 : Calculus 3

Example Question #3403 : Calculus 3

Example Question #3404 : Calculus 3

Example Question #3405 : Calculus 3

Example Question #3406 : Calculus 3

Example Question #3407 : Calculus 3

Example Question #3408 : Calculus 3

All Calculus 3 Resources

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