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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #961 : Partial Derivatives

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

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

Possible Answers:

$2z+2e^{x^2}+4x^2e^{x^2}$

2xz

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

$f_x=2xz+2xe^{x^2}$

The derivative was found using the following rules:

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

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

$f_{xz}=2x$

The rule used for finding the derivative is above.

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Example Question #962 : Partial Derivatives

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

Possible Answers:

$\small \small f_y=ze$

$\small f_y=ze^{x^2+y}$

$\small \small f_y=yze^{x^2+y}$

$\small f_y=ze^{2x+y}$

Correct answer:

$\small f_y=ze^{x^2+y}$

Explanation:

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

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

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

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Example Question #963 : Partial Derivatives

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

Possible Answers:

$\small f_z=e^{x^2+y}$

$\small f_z=2xyze^{x^2+y}$

$\small f_z=yze^{x^2+y}$

$\small f_z=ze^{x^2+y}$

Correct answer:

$\small f_z=e^{x^2+y}$

Explanation:

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

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

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

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Example Question #964 : Partial Derivatives

Find the partial derivative $\small \small f_x$ of the function $\small \small f(x,y,z)=\ln(x^2yz)$

Possible Answers:

$\small \small f_x=\frac{2}{x^2yz}$

$\small f_x=\frac{2}{x}$

$\small f_x=\frac{2}{x^2z}$

$\small f_x=\frac{2}{xyz}$

Correct answer:

$\small f_x=\frac{2}{x}$

Explanation:

To find the partial derivative $\small \small f_x$ of the function $\small \small f(x,y,z)=\ln(x^2yz)$ , we take the derivative with respect to while holding $\small \small \small y,z$ constant. So we use the chain rule to get

$\small \small f_x=\frac{\partial}{\partial x}f(x,y,z)=\frac{\partial}{\partial x}\ln(x^2yz)=\frac{1}{x^2yz}\cdot\frac{\partial}{\partial x}x^2yz$

$\small \small \small \small =\frac{2xyz}{x^2yz}=\frac{2}{x}$

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Example Question #965 : Partial Derivatives

Find the partial derivative $\small \small f_y$ of the function $\small \small f(x,y,z)=\ln(x^2yz)$ .

Possible Answers:

$\small f_y=\frac{1}{xyz}$

$\small f_y=\frac{1}{yz}$

$\small f_y=\frac{1}{x^2yz}$

$f_y=\frac{1}{y}$

Correct answer:

$f_y=\frac{1}{y}$

Explanation:

To find the partial derivative $\small \small f_y$ of the function $\small \small f(x,y,z)=\ln(x^2yz)$ , we take the derivative with respect to while holding $\small \small \small x,z$ constant. So we use the chain rule to get

$\small \small \small f_y=\frac{\partial}{\partial y}f(x,y,z)=\frac{\partial}{\partial y}\ln(x^2yz)=\frac{1}{x^2yz}\cdot\frac{\partial}{\partial y}x^2yz$

$\small \small \small \small \small =\frac{x^2z}{x^2yz}=\frac{1}{y}$

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Example Question #966 : Partial Derivatives

Find the partial derivative $\small \small f_z$ of the function $\small \small f(x,y,z)=\ln(x^2yz)$ .

Possible Answers:

$\small \small f_z=\frac{1}{x^2yz}$

$\small f_z=\frac{1}{xyz}$

$\small f_z=\frac{1}{xz}$

$\small f_z=\frac{1}{z}$

Correct answer:

$\small f_z=\frac{1}{z}$

Explanation:

To find the partial derivative $\small \small f_z$ of the function $\small \small f(x,y,z)=\ln(x^2yz)$ , we take the derivative with respect to $\small z$ while holding $\small \small \small \small \small x,y$ constant. So we use the chain rule to get

$\small \small \small \small f_z=\frac{\partial}{\partial z}f(x,y,z)=\frac{\partial}{\partial z}\ln(x^2yz)=\frac{1}{x^2yz}\cdot\frac{\partial}{\partial z}x^2yz$

$\small \small \small \small \small \small =\frac{x^2y}{x^2yz}=\frac{1}{z}$

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Example Question #967 : Partial Derivatives

Find the partial derivative $\small \small f_x$ of the function $\small \small \small f(x,y,z)=2x^2+yz+z^3$ .

Possible Answers:

$\small f_x=4x+z$

$\small f_x=4x$

$\small \small f_x=4x+3z^2$

$\small \small f_x=4x+y$

Correct answer:

$\small f_x=4x$

Explanation:

To find the partial derivative $\small \small f_x$ of the function $\small \small \small f(x,y,z)=2x^2+yz+z^3$ , we take the derivative with respect to while holding $\small \small \small \small y,z$ constant. So we get

$\small \small \small \small f_x=\frac{\partial}{\partial x}f(x,y,z)=\frac{\partial}{\partial x}(2x^2+yz+z^3)=4x$

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Example Question #968 : Partial Derivatives

Find the partial derivative $\small \small f_y$ of the function $\small \small \small f(x,y,z)=2x^2+yz+z^3$ .

Possible Answers:

f_y=y+3z^2

$\small f_y=y+z^3$

$\small f_y=z+z^3$

f_y=z

Correct answer:

f_y=z

Explanation:

To find the partial derivative $\small \small f_y$ of the function $\small \small \small f(x,y,z)=2x^2+yz+z^3$ , we take the derivative with respect to while holding $\small \small \small \small \small x,z$ constant. So we get

$\small \small \small \small \small f_y=\frac{\partial}{\partial y}f(x,y,z)=\frac{\partial}{\partial y}(2x^2+yz+z^3)=z$

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Example Question #969 : Partial Derivatives

Find the partial derivative $\small \small f_z$ of the function $\small \small \small f(x,y,z)=2x^2+yz+z^3$ .

Possible Answers:

$\small f_z=y+z^3$

f_z=y+3z^2

$\small f_z=z+3z^2$

f_z=4x+3z^2

Correct answer:

f_z=y+3z^2

Explanation:

To find the partial derivative $\small \small f_z$ of the function $\small \small \small f(x,y,z)=2x^2+yz+z^3$ , we take the derivative with respect to while holding $\small \small \small \small \small x,y$ constant. So we get

$\small \small \small \small \small \small f_z=\frac{\partial}{\partial z}f(x,y,z)=\frac{\partial}{\partial z}(2x^2+yz+z^3)=y+3z^2$

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Example Question #961 : Partial Derivatives

Find the partial derivative $\small \small \small f_x$ of the function $\small \small \small \small f(x,y,z)=x^4(y+z)^5$ .

Possible Answers:

$\small \small f_x=20x^3(y+z)^4$

$\small f_x=4x^3(y+z)^5$

$\small f_x=4x^3(y+z)^4$

$\small f_x=5x^4(y+z)^4$

Correct answer:

$\small f_x=4x^3(y+z)^5$

Explanation:

To find the partial derivative $\small \small \small f_x$ of the function $\small \small \small \small f(x,y,z)=x^4(y+z)^5$ , we take the derivative with respect to while holding $\small \small \small \small \small y,z$ constant. So we get

$\small \small f_x=\frac{\partial}{\partial x}f(x,y,z)=\frac{\partial}{\partial x}x^4(y+z)^5=(y+z)^5\frac{\partial}{\partial x}x^4=4x^3(y+z)^5$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #961 : Partial Derivatives

Example Question #962 : Partial Derivatives

Example Question #963 : Partial Derivatives

Example Question #964 : Partial Derivatives

Example Question #965 : Partial Derivatives

Example Question #966 : Partial Derivatives

Example Question #967 : Partial Derivatives

Example Question #968 : Partial Derivatives

Example Question #969 : Partial Derivatives

Example Question #961 : Partial Derivatives

All Calculus 3 Resources

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