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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #401 : Partial Derivatives

Given the function $f(x,y,z)=5e^{x+y+xyz}$ , find the partial derivative f_x(x,y,z)

Possible Answers:

$f_x(x,y,z)=5(1+yz)e^{x+y+xyz}$

$f_x(x,y,z)=5yze^{x+y+xyz}$

$f_x(x,y,z)=5(1+yz)e^{x+y+xyz}$

$f_x(x,y,z)=5(1+xz)e^{x+y+xyz}$

Correct answer:

$f_x(x,y,z)=5(1+yz)e^{x+y+xyz}$

Explanation:

We can find the partial derivative f_x(x,y,z) of the function $f(x,y,z)=5e^{x+y+xyz}$ by taking its derivative with respect to while holding and constant. We will also use the chain rule:

$f_x(x,y,z)=\frac{\partial}{\partial x}5e^{x+y+xyz}=5e^{x+y+xyz}\frac{\partial}{\partial x}(x+y+xyz)$

$=5(1+yz)e^{x+y+xyz}$

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Example Question #402 : Partial Derivatives

Given the function $f(x,y,z)=5e^{x+y+xyz}$ , find the partial derivative f_y(x,y,z)

Possible Answers:

$f_y(x,y,z)=5(1+xz)e^{x+y+xyz}$

$f_y(x,y,z)=5(1+xy)e^{x+y+xyz}$

$f_y(x,y,z)=5xze^{x+y+xyz}$

$f_y(x,y,z)=5(1+yz)e^{x+y+xyz}$

Correct answer:

$f_y(x,y,z)=5(1+xz)e^{x+y+xyz}$

Explanation:

We can find the partial derivative f_y(x,y,z) of the function $f(x,y,z)=5e^{x+y+xyz}$ by taking its derivative with respect to and holding x,z constant. We also use the chain rule to get:

$f_y(x,y,z)=\frac{\partial}{\partial y}5e^{x+y+xyz}=5e^{x+y+xyz}\frac{\partial}{\partial y}(x+y+xyz)$

$=5(1+xz)e^{x+y+xyz}$

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Example Question #401 : Partial Derivatives

Given the function $f(x,y,z)=5e^{x+y+xyz}$ , find the partial derivative f_z(x,y,z) f_y(x,y,z)

Possible Answers:

$f_z(x,y,z)=5(1+yz)e^{x+y+xyz}$

$f_z(x,y,z)=5(1+xy)e^{x+y+xyz}$

$f_z(x,y,z)=5xze^{x+y+xyz}$

$f_z(x,y,z)=5xye^{x+y+xyz}$

Correct answer:

$f_z(x,y,z)=5xye^{x+y+xyz}$

Explanation:

We can find the partial derivative f_z(x,y,z) f_y(x,y,z) of the function $f(x,y,z)=5e^{x+y+xyz}$ by taking its derivative with respect to and holding x,y constant. We also use the chain rule to get:

$f_z(x,y,z)=\frac{\partial}{\partial z}5e^{x+y+xyz}=5e^{x+y+xyz}\frac{\partial}{\partial z}(x+y+xyz)$

$=5xye^{x+y+xyz}$

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Example Question #402 : Partial Derivatives

Given the function f(x,y,z)=xy+xz+yz , find the partial derivative

f_x(x,y,z)

Possible Answers:

f_x(x,y,z)=x+y+z

f_x(x,y,z)=xy+xz

f_x(x,y,z)=y+z

f_x(x,y,z)=x+z

Correct answer:

f_x(x,y,z)=y+z

Explanation:

We can find the partial derivative f_x(x,y,z) of the function f(x,y,z)=xy+xz+yz by taking the derivative with respect to and holding and constant:

$f_x(x,y,z)=\frac{\partial}{\partial x}(xy+xz+yz)=y+z$

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Example Question #403 : Partial Derivatives

Given the function $\small f(x,y)=x^2+2xy+y^2+10$ , calculate $\small f_x(x,y)$ .

Possible Answers:

$\small \small \small f_x(x,y)=2x+2y+10$

$\small \small \small f_x(x,y)=4x$

$\small \small \small f_x(x,y)=2y+10$

$\small \small f_x(x,y)=2x+2y$

Correct answer:

$\small \small f_x(x,y)=2x+2y$

Explanation:

To find the partial derivative $\small f_x(x,y)$ of the function $\small f(x,y)=x^2+2xy+y^2+10$

we differentiate it with respect to $\small x$ while holding $\small y$ constant:

$\small \small f_x(x,y)=\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}(x^2+2xy+y^2+10)$

$\small =2x+2y$

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Example Question #404 : Partial Derivatives

Given the function $\small f(x,y)=x^2+2xy+y^2+10$ , calculate $\small \small f_y(x,y)$ .

Possible Answers:

$\small \small \small \small f_x(x,y)=2y+2x$

$\small \small \small \small f_x(x,y)=4y$

$\small \small \small \small f_x(x,y)=2x+2y^2$

$\small \small \small f_x(x,y)=2x+2y+10$

Correct answer:

$\small \small \small \small f_x(x,y)=2y+2x$

Explanation:

To find the partial derivative $\small \small f_y(x,y)$ of the function $\small f(x,y)=x^2+2xy+y^2+10$

we differentiate it with respect to $\small y$ while holding $\small x$ constant:

$\small \small \small f_y(x,y)=\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}(x^2+2xy+y^2+10)$

$\small \small \small =2y+2x$

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Example Question #405 : Partial Derivatives

If $f(x,y,z) = xyze^{xy}$ , find $f^{xz}$ .

Possible Answers:

$xye^{xy}$

None of the other answers

$ye^{xy}$

$ze^{xy}$

Correct answer:

None of the other answers

Explanation:

The correct answer is $(xy+1)ye^{xy}$ .

First we find f^z . ( f^x could be found first, it does not matter, but f^z avoids the product rule in the first step.)

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

$f^z = xye^{xy}$

Now taking the partial derivative of f^z with respect to , we obtain after using the product rule-

$f^{zx} = f^{xz} = (xy)(ye^{xy})+(y)(e^{xy}) = (xy+1)ye^{xy}$ .

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Example Question #406 : Partial Derivatives

Given $f(x,y) = xy+e^{xy}$ , find f^x .

Possible Answers:

$xy' +y + (xy'+y)e^{xy}$

None of the other answers

$y+ye^{xy}$

$y+xe^{xy}$

Correct answer:

$y+ye^{xy}$

Explanation:

To find f^x , we take the derivative of f(x,y) , using as the variable, and treating as a constant, or a number.

This means the partial derivative of with respect to is (Similar to the derivative of being , for example.)

The partial derivative of $e^{xy}$ with respect to is therefore $ye^{xy}$ , using the Chain Rule from 1-dimensional calculus.

Putting these together, we have $f^x = y+ye^{xy}$ .

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Example Question #407 : Partial Derivatives

Find the partial derivative f_x of the function f(x,y)=5xy+6x+2y .

Possible Answers:

f_x=5y+2

f_x=5y+6

f_x=6

f_x=5x+6

Correct answer:

f_x=5y+6

Explanation:

To find the partial derivative f_x of the function f(x,y)=5xy+6x+2y , we find its derivative with respect to while holding constant. We have

$f_x=\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}(5xy+6x+2y)=\frac{\partial}{\partial x}5xy+\frac{\partial}{\partial x}6x+\frac{\partial}{\partial x}2y$

=5y+6+0=5y+6

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Example Question #408 : Partial Derivatives

Find the partial derivative f_y of the function f(x,y)=5xy+6x+2y .

Possible Answers:

f_y=5x+6

f_y=2

f_y=5x+2

f_y=5y+2

Correct answer:

f_y=5x+2

Explanation:

To find the partial derivative f_y of the function f(x,y)=5xy+6x+2y , we find its derivative with respect to while holding constant. We have

$f_y=\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}(5xy+6x+2y)=\frac{\partial}{\partial y}5xy+\frac{\partial}{\partial y}6x+\frac{\partial}{\partial y}2y$

=5x+0+2=5x+2

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #401 : Partial Derivatives

Example Question #402 : Partial Derivatives

Example Question #401 : Partial Derivatives

Example Question #402 : Partial Derivatives

Example Question #403 : Partial Derivatives

Example Question #404 : Partial Derivatives

Example Question #405 : Partial Derivatives

Example Question #406 : Partial Derivatives

Example Question #407 : Partial Derivatives

Example Question #408 : Partial Derivatives

All Calculus 3 Resources

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