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Calculus 3 : Partial Derivatives

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Example Questions

Example Question #411 : Partial Derivatives

Given the function $f(x,y)=e^{\sin x+\cos y}+\pi xy$ , find the partial derivative f_x .

Possible Answers:

$f_x=-\sin y e^{\sin x+\cos y}+\pi y$

$f_x=\cos x e^{\sin x+\cos y}+\pi x$

$f_x=\sin x e^{\sin x+\cos y}+\pi y$

$f_x=\cos x e^{\sin x+\cos y}+\pi y$

Correct answer:

$f_x=\cos x e^{\sin x+\cos y}+\pi y$

Explanation:

We can find the partial derivative f_x of the function $f(x,y)=e^{\sin x+\cos y}+\pi xy$

by taking its derivative with respect to while holding constant. We use the chain rule to get

$f_x=\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}(e^{\sin x+\cos y}+\pi xy)=\frac{\partial}{\partial x}e^{\sin x+\cos y}+\frac{\partial}{\partial x}\pi xy$

$=\cos x e^{\sin x+\cos y}+\pi y$

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Example Question #412 : Partial Derivatives

Given the function $f(x,y)=e^{\sin x+\cos y}+\pi xy$ , find the partial derivative f_y .

Possible Answers:

$f_y=-\sin y e^{\sin x+\cos y}+\pi y$

$f_y=-\sin y e^{\sin x+\cos y}+\pi x$

$f_y=e^{-\sin y}+\pi x$

$f_y=\cos x e^{\sin x+\cos y}+\pi x$

Correct answer:

$f_y=-\sin y e^{\sin x+\cos y}+\pi x$

Explanation:

We can find the partial derivative f_y of the function $f(x,y)=e^{\sin x+\cos y}+\pi xy$

by taking its derivative with respect to while holding constant. We use the chain rule to get

$f_y=\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}(e^{\sin x+\cos y}+\pi xy)=\frac{\partial}{\partial y}e^{\sin x+\cos y}+\frac{\partial}{\partial y}\pi xy$

$=-\sin y e^{\sin x+\cos y}+\pi x$

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Example Question #2781 : Calculus 3

Given the function $f(x,y)=5x^2y^2+10xy+20x^3y^{3/2}$ , find the partial derivative f_x .

Possible Answers:

$f_x=10x^2y^2+10y+60x^2y^{3/2}$

$f_x=10xy^2+10x+60x^2y^{3/2}$

$f_x=5xy^2+10y+20x^3y^{3/2}$

$f_x=10xy^2+10y+60x^2y^{3/2}$

Correct answer:

$f_x=10xy^2+10y+60x^2y^{3/2}$

Explanation:

To find the partial derivative f_x of $f(x,y)=5x^2y^2+10xy+20x^3y^{3/2}$ , we take the derivative with respect to while holding constant. So we have

$f_x=\frac{\partial}{\partial x}f(x,y)=\frac{\partial}{\partial x}(5x^2y^2+10xy+20x^3y^{3/2})$

$=\frac{\partial}{\partial x}5x^2y^2+\frac{\partial}{\partial x}10xy+\frac{\partial}{\partial x}20x^3y^{3/2}$

$=10xy^2+10y+60x^2y^{3/2}$

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Example Question #2782 : Calculus 3

Find the partial derivative f_y of the function $f(x,y)=5x^2y^2+10xy+20x^3y^{3/2}$ .

Possible Answers:

$f_y=10x^2y+10y+30x^3y^{1/2}$

$f_y=10x^2y+10y+60x^3y^{1/2}$

$f_y=10x^2y^2+10x+30x^3y^{1/2}$

$f_y=10x^2y+10x+30x^3y^{1/2}$

Correct answer:

$f_y=10x^2y+10x+30x^3y^{1/2}$

Explanation:

To find the partial derivative f_y of $f(x,y)=5x^2y^2+10xy+20x^3y^{3/2}$ , we take the derivative with respect to while holding constant. So we have

$f_y=\frac{\partial}{\partial y}f(x,y)=\frac{\partial}{\partial y}(5x^2y^2+10xy+20x^3y^{3/2})$

$=\frac{\partial}{\partial y}5x^2y^2+\frac{\partial}{\partial y}10xy+\frac{\partial}{\partial y}20x^3y^{3/2}$

$=10x^2y+10x+30x^3y^{1/2}$

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Example Question #2781 : Calculus 3

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

Possible Answers:

209

225

Correct answer:

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 .

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

We find:

$f_x= 8x+0\rightarrow-16$

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Example Question #2784 : Calculus 3

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

Possible Answers:

Correct answer:

Explanation:

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

We find:

$f_x=6x^2y+4 \rightarrow72+4=76$

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Example Question #2785 : Calculus 3

Find the value of f_x for f(x,y)=ln(x^2)cos(y) at $(2,\pi)$

Possible Answers:

$-\frac{1}{2}$

$\frac{1}{2}$

Correct answer:

Explanation:

Knowledge of the following derivative rules will be necessary:

Derivative of a natural log:

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

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

Taking the partial derivative of f(x,y)=ln(x^2)cos(y) at $(2,\pi)$

We find:

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

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Example Question #2786 : Calculus 3

F(x,y)=3x^3+x^2y+3xy^2+5y^3

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

Possible Answers:

$\frac{\partial F}{\partial x}=x^2+6xy+15y^2$

$\frac{\partial F}{\partial x}=9x^2+2xy+3y^2$

$\frac{\partial F}{\partial x}=5y^3$

Correct answer:

$\frac{\partial F}{\partial x}=9x^2+2xy+3y^2$

Explanation:

$\frac{\partial F }{\partial x}$ 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 ,

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

$=\frac{\partial }{\partial x}(3x^3+x^2y+3xy^2+5y^3)$

We use the power rule on x as we take the derivative and find that

$=3*3x^{3-1}+2*x^{2-1}y+1*3x^{1-1}y^2+0*5y^3x^{0-1}$

As such

$\frac{\partial F}{\partial x}=9x^2+2xy+3y^2$

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Example Question #2787 : Calculus 3

Find the value of f_x for f(x,y)=cos(xy) at $(\pi,\frac{3}{2})$

Possible Answers:

$\pi$

$-\frac{3}{2}$

$\frac{3}{2}$

$-\pi$

Correct answer:

$\frac{3}{2}$

Explanation:

Knowledge of the following derivative rules will be necessary:

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)=cos(xy) at $(\pi,\frac{3}{2})$

We find:

$f_x= -ysin(xy)\rightarrow\frac{3}{2}$

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Example Question #2788 : Calculus 3

Given f(x,y,z,w) = xyzw+(xyzw)^2 , find $f_{xw}$

Possible Answers:

None of the other answers

2xyzw

yz+4xy^2z^2w

xw+4xw

yz+y^2z^2

Correct answer:

yz+4xy^2z^2w

Explanation:

To find $f_{xw}$ , first we take the partial derivative with respect to , keeping the other variables constant (i.e. treating the other variables as numbers)

So the partial derivative of with respect to is

f_x = yzw +2(xyzw)yzw =yzw +2xy^2z^2w^2 .

Now we take the partial derivative of f_x with respect to , keeping the other variables constant.

$f_{xw} = yz+4xy^2z^2w$ .

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Calculus 3 : Partial Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #411 : Partial Derivatives

Example Question #412 : Partial Derivatives

Example Question #2781 : Calculus 3

Example Question #2782 : Calculus 3

Example Question #2781 : Calculus 3

Example Question #2784 : Calculus 3

Example Question #2785 : Calculus 3

Example Question #2786 : Calculus 3

Example Question #2787 : Calculus 3

Example Question #2788 : Calculus 3

All Calculus 3 Resources

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