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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #951 : Partial Derivatives

Find the partial derivative f_x of the function $f(x,y,z)=ze^{x^2+y}$ .

Possible Answers:

$f_x=e^{x^2+y}$

$f_x=ze^{2x}$

$f_x=yze^{x^2+y}$

$f_x=2xze^{x^2+y}$

Correct answer:

$f_x=2xze^{x^2+y}$

Explanation:

To find the partial derivative f_x of the function $f(x,y,z)=ze^{x^2+y}$ , we take the derivative with respect to $\small x$ while holding $\small y,z$ constant. We also use the chain rule to get

$\small \small \small f_x=\frac{\partial}{\partial x}f(x,y,z)=\frac{\partial}{\partial x}ze^{x^2+y}=ze^{x^2+y}\frac{\partial}{\partial x}(x^2+y)$

$\small =2xze^{x^2+y}$

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Example Question #952 : Partial Derivatives

Evaluate for $\frac{\partial F}{\partial x}$ .

F(x,y)=xy^2+4x^2y+3x

Possible Answers:

$\frac{\partial F}{\partial x}=y^2+8xy+3$

$\frac{\partial F}{\partial x}=2xy+4x^2$

$\frac{\partial F}{\partial x}=3y^2+8y$

$\frac{\partial F}{\partial x}=xy+4x^2$

Correct answer:

$\frac{\partial F}{\partial x}=y^2+8xy+3$

Explanation:

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

Taking the partial derivative of both sides

$\frac{\partial }{\partial x}[F(x,y)]=\frac{\partial }{\partial x}[xy^2+4x^2y+3x]$

yields partial partial

$\frac{\partial F}{\partial x}=y^2+8xy+3$

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Example Question #953 : Partial Derivatives

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

$F(x,y)=e^xtan\,y$

Possible Answers:

$\frac{\partial F}{\partial y}=e^xsec^2y+e^xtan\,y$

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

$\frac{\partial F}{\partial y}=e^xsec^2y$

Correct answer:

$\frac{\partial F}{\partial y}=e^xsec^2y$

Explanation:

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

Taking the partial derivative of both sides

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

yields partial partial

$\frac{\partial F}{\partial y}=e^xsec^2y$

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Example Question #954 : Partial Derivatives

Solve for the partial derivative.

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

Possible Answers:

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

$\frac{\partial F}{\partial x}=sec^2x-csc^2y$

$\frac{\partial F}{\partial x}=-csc^2x$

Correct answer:

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

Explanation:

We are solving for the partial derivative with respect to . Doing so we treat the other variables as constants.

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

We then find that

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

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Example Question #955 : Partial Derivatives

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

Possible Answers:

2xy

3x^2

Correct answer:

Explanation:

To find the partial derivative of the function, you must take a total of three partial derivatives consecutively: $\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}$ . $\frac{\partial }{\partial x}(x^3y+xz^3)=3x^2y+z^3$ . $\frac{\partial }{\partial y}(3x^2y+z^2)=3x^2$ . $\frac{\partial }{\partial z}(3x^2)=6x$ .

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Example Question #956 : Partial Derivatives

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

$f(x, y, z)=\cos(z^2+z)-\sin(xyz)$

Possible Answers:

$-(2z+1)\sin(z^2+z)-xy\cos(xyz)$

$(2z+1)\sin(z^2+z)-xy\cos(xyz)$

$-(2z+1)\sin(z^2+z)+xy\cos(xyz)$

$-(2z+1)\sin(z^2+z)-z\cos(xyz)$

Correct answer:

$-(2z+1)\sin(z^2+z)-xy\cos(xyz)$

Explanation:

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

Taking the partial derivative of the function with respect to z, we get

$f_z=-(2z+1)\sin(z^2+z)-xy\cos(xyz)$

The following rules were used to find the derivative:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sin(x)=\cos(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

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Example Question #957 : Partial Derivatives

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

Possible Answers:

$\cos(x)+6yz$

$\cos(x)+3xyz$

$-\sin(x)+6xyz$

$-\cos(x) +6yz$

Correct answer:

$-\cos(x) +6yz$

Explanation:

To solve, you must do a total of three partial derivatives consecutively: $\frac{\partial }{\partial x},\frac{\partial }{\partial x},\frac{\partial }{\partial x}$ . Using the rules for partial derivatives, we get $f_x=\cos(x)+3x^2yz$ , $f_{xx}=-\sin(x)+6xyz$ , and finally, $f_{xxx}=-\cos(x)+6yz$

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Example Question #958 : Partial Derivatives

Find f_y of the following function:

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

Possible Answers:

$2y\sec(z)-x\sin(xy)$

$2y\sec(z)+x\sin(xy)$

$2y+x\sin(xy)$

Correct answer:

$2y\sec(z)+x\sin(xy)$

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 y is

$f_y=2y\sec(z)+x\sin(xy)$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

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Example Question #959 : Partial Derivatives

Find f_y of the following function:

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

Possible Answers:

$y+y\sec^2(xy)$

$x+x\sec^2(xy)$

$x-x\sec^2(xy)$

$x\tan^2(xy)$

Correct answer:

$x+x\sec^2(xy)$

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 y is

$f_y=x+x\sec^2(xy)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

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Example Question #960 : Partial Derivatives

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

$f(x, y, z)=z^3e^{xy}$

Possible Answers:

$6xze^{xy}$

$6ze^{xy}$

$3z^2e^{xy}$

Correct answer:

$6xze^{xy}$

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

$f_z=3z^2e^{xy}$

The derivative was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x}ax=a$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

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

$f_{zz}=6ze^{xy}$

The derivative was found using the same rules as above.

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

$f_{zzy}=6xze^{xy}$

The derivative was found using the above rules along with

$\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u \frac{\mathrm{d} u}{\mathrm{d} x}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #951 : Partial Derivatives

Example Question #952 : Partial Derivatives

Example Question #953 : Partial Derivatives

Example Question #954 : Partial Derivatives

Example Question #955 : Partial Derivatives

Example Question #956 : Partial Derivatives

Example Question #957 : Partial Derivatives

Example Question #958 : Partial Derivatives

Example Question #959 : Partial Derivatives

Example Question #960 : Partial Derivatives

All Calculus 3 Resources

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