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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #641 : Partial Derivatives

Find the partial derivative $\small f_y$ of $\small f(x,y)=2xy+5\pi x^{3/2}$

Possible Answers:

$\small \small f_y=2x$

$\small \small \small f_y=2y+\frac{15\pi}{2}\sqrt{x}$

$\small \small \small \small f_y=2y$

$\small \small \small \small \small f_y=2x+5\pi x^{3/2}$

Correct answer:

$\small \small f_y=2x$

Explanation:

To find the partial derivative $\small f_y$ of $\small f(x,y)=2xy+5\pi x^{3/2}$ , we need to take its derivative with respect to $\small y$ , which means holding $\small x$ constant. So we get

$\small \small \small f_y =\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}[2xy+5\pi x^{3/2}]=\frac{\partial}{\partial y}2xy+\frac{\partial}{\partial y}5\pi x^{3/2}$

$\small \small =2x+0=2x$

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Example Question #642 : Partial Derivatives

Find the partial derivative $\small \small f_x$ of $\small \small \small f(x,y)=e^{5x^5+2y^{3/2}}$

Possible Answers:

$\small \small \small \small \small f_x=3\sqrt{y}e^{5x^5+2y^{3/2}}$

$\small \small \small \small f_x=25x^4e^{5x^5+2y^{3/2}}$

$\small \small \small \small \small f_x=5x^4e^{25x^4+2y^{3/2}}$

$\small \small \small \small \small f_x=25x^4e^{25x^4}$

Correct answer:

$\small \small \small \small f_x=25x^4e^{5x^5+2y^{3/2}}$

Explanation:

To find the partial derivative $\small \small f_x$ of $\small \small \small f(x,y)=e^{5x^5+2y^{3/2}}$ , we need to take its derivative with respect to $\small x$ while holding constant. We use the chain rule to get

$\small \small \small \small f_x=\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}e^{5x^5+2y^{3/2}}= e^{5x^5+2y^{3/2}}\cdot \frac{\partial}{\partial x}(5x^5+2y^{3/2})$

$\small \small \small \small \small =25x^4e^{5x^5+2y^{3/2}}$

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Example Question #643 : Partial Derivatives

Find the partial derivative $\small \small \small f_y$ of $\small \small \small f(x,y)=e^{5x^5+2y^{3/2}}$

Possible Answers:

$\small \small f_y=3\sqrt{y}e^{5x^5+2y^{3/2}}$

$\small \small \small f_y=25x^4e^{5x^5+2y^{3/2}}$

$\small \small \small \small f_y=e^{3\sqrt{y}}$

$\small \small \small f_y=2y^{3/2}e^{5x^5+3\sqrt{y}}$

Correct answer:

$\small \small f_y=3\sqrt{y}e^{5x^5+2y^{3/2}}$

Explanation:

To find the partial derivative $\small \small \small f_y$ of $\small \small \small f(x,y)=e^{5x^5+2y^{3/2}}$ , we need to take its derivative with respect to $\small y$ while holding $\small x$ constant. We use the chain rule to get

$\small \small \small \small \small f_y=\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}e^{5x^5+2y^{3/2}}= e^{5x^5+2y^{3/2}}\cdot \frac{\partial}{\partial y}(5x^5+2y^{3/2})$

$\small \small \small \small \small \small \small =\frac{3}{2}2y^{1/2}e^{5x^5+2y^{3/2}}=3\sqrt{y}e^{5x^5+2y^{3/2}}$

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Example Question #644 : Partial Derivatives

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

Possible Answers:

$\small \small f_x=\ln(3x^2)$

$\small f_x=\frac{3x^2}{x^3+2y^2}$

$\small \small f_x=\frac{3x^2}{3x^2+4y}$

$\small \small f_x=\frac{4y}{x^3+2y^2}$

Correct answer:

$\small f_x=\frac{3x^2}{x^3+2y^2}$

Explanation:

To find the partial derivative $\small \small \small f_x$ of $\small f(x,y)=\ln(x^3+2y^2)$ , we need to take its derivative with respect to while holding constant. We use that chain rule to get

$\small \small f_x=\frac{\partial}{\partial x} f(x,y)=\frac{\partial}{\partial x}\ln(x^3+2y^2)=\frac{1}{x^3+2y^2}\frac{\partial}{\partial x}(x^3+2y^2)$

$\small =\frac{3x^2}{x^3+2y^2}$

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Example Question #645 : Partial Derivatives

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

Possible Answers:

$\small \small \small \small f_y=\frac{4y}{3x^2+4y}$

$\small \small \small \small f_y=\frac{3x^2}{x^3+2y^2}$

$\small \small \small \small f_y=\frac{4y}{x^3+2y^2}$

$\small \small \small \small f_y=\ln(4y)$

Correct answer:

$\small \small \small \small f_y=\frac{4y}{x^3+2y^2}$

Explanation:

To find the partial derivative $\small \small \small \small f_y$ of $\small f(x,y)=\ln(x^3+2y^2)$ , we need to take its derivative with respect to $\small y$ while holding $\small x$ constant. We use that chain rule to get

$\small \small \small f_y=\frac{\partial}{\partial y} f(x,y)=\frac{\partial}{\partial y}\ln(x^3+2y^2)=\frac{1}{x^3+2y^2}\frac{\partial}{\partial y}(x^3+2y^2)$

$\small \small =\frac{4y}{x^3+2y^2}$

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Example Question #646 : Partial Derivatives

Find the partial derivative $\small \small \small \small \small f_x$ of $\small \small f(x,y,z)=x^5y^{10}z^{15}$

Possible Answers:

$\small \small \small f_x=5x^4y^{10}z^{15}$

$\small \small \small \small f_x=0$

$\small \small \small \small f_x=5x^5y^{10}z^{15}$

$\small \small \small \small f_x=50x^4y^{9}z^{15}$

Correct answer:

$\small \small \small f_x=5x^4y^{10}z^{15}$

Explanation:

To find the partial derivative $\small \small \small \small \small f_x$ of $\small \small f(x,y,z)=x^5y^{10}z^{15}$ , we need to take its derivative with respect to $\small x$ while holding $\small y,z$ constant. So we have

$\small \small \small f_x=\frac{\partial}{\partial x} f(x,y,z)=\frac{\partial}{\partial x} x^5y^{10}z^{15}=y^{10}z^{15}\frac{\partial}{\partial x}x^5$

$\small =5x^4y^{10}z^{15}$

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Example Question #647 : Partial Derivatives

Find the partial derivative $\small \small \small \small \small \small f_y$ of $\small \small f(x,y,z)=x^5y^{10}z^{15}$

Possible Answers:

$\small \small \small \small f_y=10x^5y^{9}z^{15}$

$\small \small \small \small f_y=x^4y^{9}z^{15}$

$\small \small \small \small f_y=0$

$\small \small \small \small f_y=50x^4y^{9}z^{15}$

Correct answer:

$\small \small \small \small f_y=10x^5y^{9}z^{15}$

Explanation:

To find the partial derivative $\small \small \small \small \small \small f_y$ of $\small \small f(x,y,z)=x^5y^{10}z^{15}$ , we need to take its derivative with respect to $\small y$ while holding $\small \small x,z$ constant. So we have

$\small \small \small \small f_y=\frac{\partial}{\partial y} f(x,y,z)=\frac{\partial}{\partial y} x^5y^{10}z^{15}=x^{5}z^{15}\frac{\partial}{\partial x}y^{10}$

$\small \small \small =10x^5y^{9}z^{15}$

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Example Question #648 : Partial Derivatives

Find the partial derivative $\small \small \small \small \small \small \small f_z$ of $\small \small f(x,y,z)=x^5y^{10}z^{15}$

Possible Answers:

$\small \small \small \small \small \small f_z=15x^5y^{10}z^{13}$

$\small \small \small \small \small f_z=15x^5y^{10}z^{14}$

$\small \small \small \small \small \small f_z=150x^4y^{9}z^{14}$

$\small \small \small \small \small \small f_z=150x^5y^{9}z^{14}$

Correct answer:

$\small \small \small \small \small f_z=15x^5y^{10}z^{14}$

Explanation:

To find the partial derivative $\small \small \small \small \small \small \small f_z$ of $\small \small f(x,y,z)=x^5y^{10}z^{15}$ , we need to take its derivative with respect to $\small z$ while holding $\small \small \small x,y$ constant. So we have

$\small \small \small \small \small f_z=\frac{\partial}{\partial z} f(x,y,z)=\frac{\partial}{\partial z} x^5y^{10}z^{15}=x^{5}y^{10}\frac{\partial}{\partial x}z^{15}$

$\small \small \small \small =15x^5y^{10}z^{14}$

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Example Question #649 : Partial Derivatives

Find the partial derivative $\small \small \small \small \small \small \small \small \small f_y$ of $\small \small \small \small f(x,y,z)=e^{x^2y^{3}z^{10}}$

Possible Answers:

$\small \small \small \small \small \small \small \small f_y=0$

$\small \small \small \small \small \small \small \small f_y=3y^2z^{10}e^{x^2y^3z^{10}}$

$\small \small \small \small \small \small \small \small f_y=3y^2e^{x^2y^3z^{10}}$

$\small \small \small \small \small \small \small f_y=3x^2y^2z^{10}e^{x^2y^3z^{10}}$

Correct answer:

$\small \small \small \small \small \small \small f_y=3x^2y^2z^{10}e^{x^2y^3z^{10}}$

Explanation:

To find the partial derivative $\small \small \small \small \small \small \small \small \small f_y$ of $\small \small \small \small f(x,y,z)=e^{x^2y^{3}z^{10}}$ , we need to take its derivative with respect to $\small y$ while holding $\small \small \small \small \small x,z$ constant. So we use the chain rule to get

$\small \small \small \small \small \small \small f_y=\frac{\partial}{\partial y} f(x,y,z)=\frac{\partial}{\partial y} e^{x^2y^{3}z^{10}}=e^{x^2y^3z^{10}}\frac{\partial}{\partial y}x^2y^3z^{10}$

$\small \small \small \small \small \small =3x^2y^2z^{10}e^{x^2y^3z^{10}}$

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Example Question #650 : Partial Derivatives

Find the partial derivative $\small \small \small \small \small \small \small \small \small \small f_z$ of $\small \small \small \small f(x,y,z)=e^{x^2y^{3}z^{10}}$

Possible Answers:

$\small \small \small \small \small \small \small \small \small f_z=z^{10}e^{x^2y^3z^{10}}$

$\small \small \small \small \small \small \small \small \small f_z=10z^{9}e^{x^2y^3z^{10}}$

$\small \small \small \small \small \small \small \small f_z=10x^2y^3z^{9}e^{x^2y^3z^{10}}$

$\small \small \small \small \small \small \small \small \small f_z=e^{10x^2y^3z^9}$

Correct answer:

$\small \small \small \small \small \small \small \small f_z=10x^2y^3z^{9}e^{x^2y^3z^{10}}$

Explanation:

To find the partial derivative $\small \small \small \small \small \small \small \small \small \small f_z$ of $\small \small \small \small f(x,y,z)=e^{x^2y^{3}z^{10}}$ , we need to take its derivative with respect to $\small z$ while holding $\small \small \small \small \small \small x,y$ constant. So we use the chain rule to get

$\small \small \small \small \small \small \small \small f_z=\frac{\partial}{\partial z} f(x,y,z)=\frac{\partial}{\partial z} e^{x^2y^{3}z^{10}}=e^{x^2y^3z^{10}}\frac{\partial}{\partial z}x^2y^3z^{10}$

$\small \small \small \small \small \small \small =10x^2y^3z^{9}e^{x^2y^3z^{10}}$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #641 : Partial Derivatives

Example Question #642 : Partial Derivatives

Example Question #643 : Partial Derivatives

Example Question #644 : Partial Derivatives

Example Question #645 : Partial Derivatives

Example Question #646 : Partial Derivatives

Example Question #647 : Partial Derivatives

Example Question #648 : Partial Derivatives

Example Question #649 : Partial Derivatives

Example Question #650 : Partial Derivatives

All Calculus 3 Resources

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