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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #2781 : Calculus 3

F(x,y)=cos (x) + sin (y) -x^2+xy-y^2

Find $\frac{\partial F}{\partial y}$

Possible Answers:

$\frac{\partial F}{\partial y}=-sin(x)-2x+y$

$\frac{\partial F}{\partial y}=cos (x) -x^2$

$\frac{\partial F}{\partial y}=cos (y) +x-2y$

Correct answer:

$\frac{\partial F}{\partial y}=cos (y) +x-2y$

Explanation:

$\frac{\partial F }{\partial y}$ is defined as the partial derivative of taken with respect to . Any other variable is treated as a constant.

Finding the partial derivative with respect to ,

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

$=\frac{\partial }{\partial y}(cos (x) + sin (y) -x^2+xy-y^2)$

We use the power rule on and the definition of trigonometric derivatives as we take the derivative and find that

$=0*cos(x)y^{0-1}+cos(y)-0*x^2y^{0-1}+1*xy^{1-1}-2*y^{2-1}$

As such

$\frac{\partial F}{\partial y}=cos (y) +x-2y$

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Example Question #2787 : Calculus 3

Find the value of f_x for $f(x,y)=e^{x^2y}+5x$ at (2,2)

Possible Answers:

4e^4+5

8e^8+5

4e^8+5

e^4+5

8e^4+5

Correct answer:

8e^8+5

Explanation:

Note that for this problem, we're told to take the derivative with respect to one particular variable. This is known as taking a partial derivative; often it is denoted with the Greek character delta, $\delta$ , or by the subscript of the variable being considered such as f_x or f_y .

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential:

d[e^u]=e^udu

Note that u may represent large functions, and not just individual variables!

Taking the partial derivative of $f(x,y)=e^{x^2y}+5x$ at (2,2)

We find:

$f_x=2xye^{x^2y}+5 \rightarrow8e^8+5$

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Example Question #421 : Partial Derivatives

Find the value of f_x for f(x,y)=x^2y+3xy+4y^2 at (5,-4)

Possible Answers:

Correct answer:

Explanation:

Taking the partial derivative of f(x,y)=x^2y+3xy+4y^2 at (5,-4)

We find:

$f_x= 2xy+3y+0\rightarrow-40-12=-52$

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Example Question #422 : Partial Derivatives

Find the value of f_x for f(x,y)=tan(xy) at $(3,\pi)$

Possible Answers:

$-\pi$

$3\pi$

$-3\pi$

$\pi$

Correct answer:

$\pi$

Explanation:

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative:

$d[tan(u)]=\frac{du}{cos^2(u)}$

Note that u may represent large functions, and not just individual variables!

Taking the partial derivative of f(x,y)=tan(xy) at $(3,\pi)$

We find:

$f_x=\frac{y}{cos^2(xy)} \rightarrow\pi$

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Example Question #423 : Partial Derivatives

Find the value of f_x for f(x,y)=sin^2(xy) at $(\pi,\frac{1}{4})$

Possible Answers:

$\frac{1}{4}$

$-\frac{1}{2}$

$\frac{\pi}{2}$

$\frac{1}{2}$

$-\frac{\pi}{2}$

Correct answer:

$\frac{1}{4}$

Explanation:

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative:

d[sin(u)]=cos(u)du

Product rule: d[uv]=udv+vdu

Note that u may represent large functions, and not just individual variables!

Taking the partial derivative of f(x,y)=sin^2(xy) at $(\pi,\frac{1}{4})$

We find:

$f_x= 2ysin(xy)cos(xy)\rightarrow\frac{1}{2}sin(\frac{\pi}{4})cos(\frac{\pi}{4})=\frac{1}{4}$

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Example Question #424 : Partial Derivatives

Find the value of f_x for f(x,y)=sin(2x)tan(y) at $(2\pi,\frac{\pi}{4})$

Possible Answers:

Correct answer:

Explanation:

Knowledge of the following derivative rules will be necessary:

Trigonometric derivative:

d[sin(u)]=cos(u)du

Note that u may represent large functions, and not just individual variables!

Taking the partial derivative of f(x,y)=sin(2x)tan(y) at $(2\pi,\frac{\pi}{4})$

We find:

$f_x= 2cos(2x)tan(y)\rightarrow2cos(2(2\pi))tan(\frac{\pi}{4})=2$

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Example Question #425 : Partial Derivatives

Find the value of f_x for f(x,y)=7x^2y^3 at (3,-2)

Possible Answers:

168

504

Correct answer:

Explanation:

Taking the partial derivative of f(x,y)=7x^2y^3 at (3,-2)

We find:

$f_x= 14xy^3\rightarrow -336$

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Example Question #426 : Partial Derivatives

Find the value of f_x for f(x,y)=sin(yln(x)) at $(1,\frac{1}{2})$

Possible Answers:

$\frac{1}{2}$

$\pi$

Correct answer:

$\frac{1}{2}$

Explanation:

Knowledge of the following derivative rules will be necessary:

Derivative of a natural log:

$d[ln(u)]=\frac{du}{u}$

Trigonometric derivative:

d[sin(u)]=cos(u)du

Note that u may represent large functions, and not just individual variables!

Taking the partial derivative of f(x,y)=sin(yln(x)) at $(1,\frac{1}{2})$

We find:

$f_x= \frac{y}{x}cos(yln(x))\rightarrow \frac{1}{2}$

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Example Question #427 : Partial Derivatives

Find the value of f_x for f(x,y)=ln(cos(xy)) at $(\frac{1}{4},\pi)$

Possible Answers:

$\frac{1}{4}$

$\pi$

$-\pi$

$-\frac{1}{4}$

Correct answer:

$-\pi$

Explanation:

Knowledge of the following derivative rules will be necessary:

Derivative of a natural log:

$d[ln(u)]=\frac{du}{u}$

Trigonometric derivative:

d[cos(u)]=-sin(u)du

Note that u may represent large functions, and not just individual variables!

Taking the partial derivative of f(x,y)=ln(cos(xy)) at $(\frac{1}{4},\pi)$

We find:

$f_x=\frac{-ysin(xy)}{cos(xy)} \rightarrow \frac{-\pi sin(\frac{\pi}{4})}{cos(\frac{\pi}{4})}=-\pi$

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Example Question #428 : Partial Derivatives

Given the function $f(x,y,z)=\sin(x^2+y^3+z^5)$ , find the partial derivative f_x .

Possible Answers:

$f_x=3x^2\cos(x^2+y^3+z^5)$

$f_x=3y^2\cos(x^2+y^3+z^5)$

$f_x=2x\sin(x^2+y^3+z^5)$

$f_x=2x\cos(x^2+y^3+z^5)$

Correct answer:

$f_x=2x\cos(x^2+y^3+z^5)$

Explanation:

Given the function

$f(x,y,z)=\sin(x^2+y^3+z^5)$ ,

we find f_x by taking the derivative of

$f(x,y,z)=\sin(x^2+y^3+z^5)$

with respect to , which means we hold y,z constant.

So we get (using the chain rule)

$\\f_x=\frac{\partial}{\partial x}f(x,y,z) \\f_x=\frac{\partial}{\partial x}\sin(x^2+y^3+z^5) \\f_x=\cos(x^2+y^3+z^5)\cdot \frac{\partial}{\partial x}(x^2+y^3+z^5)$

$=2x\cos(x^2+y^3+z^5)$

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #2781 : Calculus 3

Example Question #2787 : Calculus 3

Example Question #421 : Partial Derivatives

Example Question #422 : Partial Derivatives

Example Question #423 : Partial Derivatives

Example Question #424 : Partial Derivatives

Example Question #425 : Partial Derivatives

Example Question #426 : Partial Derivatives

Example Question #427 : Partial Derivatives

Example Question #428 : Partial Derivatives

All Calculus 3 Resources

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