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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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:

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

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

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

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

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:

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

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

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

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

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:

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

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

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

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

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 #961 : 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:

$\small f_x=4x+z$

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

$\small f_x=4x$

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

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 #3331 : Calculus 3

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=z

$\small f_y=y+z^3$

f_y=y+3z^2

$\small f_y=z+z^3$

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:

f_z=y+3z^2

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

$\small f_z=y+z^3$

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 #970 : 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:

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

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

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

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

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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Example Question #971 : Partial Derivatives

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

Possible Answers:

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

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

$\small \small \small f_y=\frac{1}{6}x^4(y+z)^6$

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

Correct answer:

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

Explanation:

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

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

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

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Example Question #972 : Partial Derivatives

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

To find the partial derivative $\small \small \small \small f_z$ 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 $\small \small \small \small \small x,y$ constant. So we use the power rule to get

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

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

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #963 : Partial Derivatives

Example Question #964 : Partial Derivatives

Example Question #965 : Partial Derivatives

Example Question #961 : Partial Derivatives

Example Question #967 : Partial Derivatives

Example Question #3331 : Calculus 3

Example Question #969 : Partial Derivatives

Example Question #970 : Partial Derivatives

Example Question #971 : Partial Derivatives

Example Question #972 : Partial Derivatives

All Calculus 3 Resources

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