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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #3409 : Calculus 3

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

f(x,y,z)=6y^2e^z+xz^2

Possible Answers:

6y^2e^z+2x

e^z+2x

6y^2e^z+2xz

Correct answer:

6y^2e^z+2x

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 z:

f_z=6y^2e^z+2xz

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

$f_{zz}=6y^2e^z+2x$

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Example Question #3410 : Calculus 3

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

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

Possible Answers:

$-z^2\cos(xz)+2y^2e^z$

4ye^z

4e^z

$-z\sin(xz)+2xye^z$

Correct answer:

4ye^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 x:

$f_x=-z\sin(xz)+2xy^2e^z$

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

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

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

$f_{xxy}=4ye^z$

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

$f_{xxyz}=4ye^z$

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Example Question #1041 : Partial Derivatives

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

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

Possible Answers:

$6x+y^2\sec(xy)\tan(xy)$

$6x+2y^3\sec(xy)\tan(xy)$

$3x^2+2y^3\sec(xy)\tan(xy)$

$6x+y^3\sec(xy)\tan(xy)$

$6x+2y^2\sec(xy)\tan(xy)$

Correct answer:

$6x+2y^3\sec(xy)\tan(xy)$

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.

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

$f_x=3x^2+y^2\sec^2(xy)$

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

$f_{xx}=6x+2y^3\sec(xy)\tan(xy)$

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Example Question #1042 : Partial Derivatives

Find $f_{xxz}$ of the function $x^2z^2+x^3\cos(z)$

Possible Answers:

$4z-6x\sin(z)$

$4z-x\cos(z)$

$4z^2-6x\sin(z)$

$4z+6x\sin(z)$

Correct answer:

$4z-6x\sin(z)$

Explanation:

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

$\frac{\partial }{\partial x}(x^2z^2+x^3\cos(z))=2xz^2+3x^2\cos(z)$

$\frac{\partial }{\partial x}(2xz^2+3x^2\cos(z))=2z^2+6x\cos(z)$

$\frac{\partial }{\partial z}(2z^2+6x\cos(z))=4z-6x\sin(z)$

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Example Question #1043 : Partial Derivatives

Find $f_{xxz}$ of the function $5x^2z^5+ze^{5x}$

Possible Answers:

$50z^4-25e^{10x}$

$50z^4+25e^{5x}$

$50z^3+20e^{5x}$

$50z^5+20e^{4x}$

Correct answer:

$50z^4+25e^{5x}$

Explanation:

To find $f_{xxz}$ of the function, you must take three consecutive partial derivatives:

$\frac{\partial }{\partial x}(5x^2z^5+ze^{5x})=10xz^5+5ze^{5x}$

$\frac{\partial }{\partial x}(10xz^5+5ze^{5x})=10z^5+25ze^{5x}$

$\frac{\partial }{\partial z}(10z^5+25ze^{5x})=50z^4+25e^{5x}$

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Example Question #1044 : Partial Derivatives

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

$F(x,y)=e^x+e^y+cos\,x+sin\,y$

Possible Answers:

$\frac{\partial F}{\partial y}=e^{xy}+cos\,y-sin\,x$

$\frac{\partial F}{\partial y}=e^y+cos\,y$

$\frac{\partial F}{\partial y}=e^x-sin\,x$

Correct answer:

$\frac{\partial F}{\partial y}=e^y+cos\,y$

Explanation:

To find $\frac{\partial F}{\partial y}$ we take the derivative of the function F(x,y) with respect to and treat the other variables as constants.

As such,

$\frac{\partial }{\partial y}[F(x,y)]=\frac{\partial }{\partial y}[e^x+e^y+cos\,x+sin\,y]$

yields

$\frac{\partial F}{\partial y}=e^y+cos\,y$

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Example Question #1045 : Partial Derivatives

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

$F(x,y)=tan\,x-cot\,y+y^2$

Possible Answers:

$\frac{\partial F}{\partial x}=0$

$\frac{\partial F}{\partial x}=sec^2\,x$

$\frac{\partial F}{\partial x}=csc^2\,y+2y$

Correct answer:

$\frac{\partial F}{\partial x}=sec^2\,x$

Explanation:

To find $\frac{\partial F}{\partial x}$ we take the derivative of the function F(x,y) with respect to and treat the other variables as constants.

As such,

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

yields

$\frac{\partial F}{\partial x}=sec^2\,x$

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Example Question #1046 : Partial Derivatives

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

$F(x,y)=sec\,x+csc\,y$

Possible Answers:

$\frac{\partial F}{\partial x}=sec(x)tan(x)$

$\frac{\partial F}{\partial x}=sec^2\,x$

$\frac{\partial F}{\partial x}=-csc(x)cot(x)$

Correct answer:

$\frac{\partial F}{\partial x}=sec(x)tan(x)$

Explanation:

To find $\frac{\partial F}{\partial x}$ we take the derivative of the function F(x,y) with respect to and treat the other variables as constants.

As such,

$\frac{\partial }{\partial x}[F(x,y)]=\frac{\partial }{\partial x}[sec\,x+csc\,y]$

yields

$\frac{\partial F}{\partial x}=sec(x)tan(x)$

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Example Question #1047 : Partial Derivatives

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

f(x,y,z)=z^3y^4+xz^2y

Possible Answers:

12y^3z^2+2xz

72y^2z

36y^2z^2

36y^2z

Correct answer:

72y^2z

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

f_y=4y^3z^3+xz^2

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

$f_{yz}=12y^3z^2+2xz$

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

$f_{yzy}=36y^2z^2$

Finally, we take the partial derivative of this function with respect to z:

$f_{yzyz}=72y^2z$

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Example Question #1041 : Partial Derivatives

Find $f_{xy}$ of the function $x^3\sin(y)+xyz$

Possible Answers:

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

$-3x^2\cos(y)+z$

$3x^2\cos(y)-z$

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

Correct answer:

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

Explanation:

To find $f_{xy}$ of the function, you must take two consecutive partial derivatives:

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

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

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #3409 : Calculus 3

Example Question #3410 : Calculus 3

Example Question #1041 : Partial Derivatives

Example Question #1042 : Partial Derivatives

Example Question #1043 : Partial Derivatives

Example Question #1044 : Partial Derivatives

Example Question #1045 : Partial Derivatives

Example Question #1046 : Partial Derivatives

Example Question #1047 : Partial Derivatives

Example Question #1041 : Partial Derivatives

All Calculus 3 Resources

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