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Calculus 1 : How to find differential functions

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Example Questions

Example Question #438 : Functions

The magnitude of a particle's noise varies with position and is given by the function

$f(x,y,z)=x^2ye^{xz^2}\sin(y+z)$

Describe rate of the change in magnitude as the value of x changes.

Possible Answers:

$xye^{xz^2}\sin(y+z)(2+xz^2)$

$xye^{xz^2}\cos(y+z)(2+xz^2)$

$xze^{xz^2}\sin(y+z)(2+xy^2)$

$xze^{xz^2}\cos(y+z)(2+xy^2)$

$xy^2e^{xz^2}\sin(y+z)(2+xz^2)$

Correct answer:

$xye^{xz^2}\sin(y+z)(2+xz^2)$

Explanation:

Rates of change of a function can be found by taking the derivative of the function. Note that a rate of change can depend on multiple variables, and sometimes we only wish to know the influence of one variable

For this problem, we're being asked to take the derivative with respect to one particular variable, in this case x. 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

Product rule: d[uv]=udv+vdu

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

The function we're given is

$f(x,y,z)=x^2ye^{xz^2}\sin(y+z)$

We'll be taking the derivative with respect to x using the above rules:

$\frac{\delta f(x,y,z)}{\delta x}=2xye^{xz^2}\sin(y+z)+x^2yz^2e^{xz^2}\sin(y+z)$

$\frac{\delta f(x,y,z)}{\delta x}=xye^{xz^2}\sin(y+z)(2+xz^2)$

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Example Question #251 : How To Find Differential Functions

Take the derivative with respect to y of the function $f(x,y,z)=x^2yz^3\cos(xy^2z)$

Possible Answers:

$x^4z^2(\cos(xy^2z)-2xy^2z\sin(xy^2z))$

$xz(\cos(xy^2z)+xy^2z\sin(xy^2z))$

$x^2z^3(\cos(xy^2z)-2xy^2z\sin(xy^2z))$

$x^2z(\cos(xy^2z)-2xy^4z\sin(xy^2z))$

$x^3z^3(\cos(xy^2z)-2xyz\sin(xy^2z))$

Correct answer:

$x^2z^3(\cos(xy^2z)-2xy^2z\sin(xy^2z))$

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:

Trigonometric derivative: $d[\cos(u)]=-\sin(u)du$

Product rule: d[uv]=udv+vdu

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

For the function

$f(x,y,z)=x^2yz^3\cos(xy^2z)$

$\frac{\delta f(x,y,z)}{\delta y} = x^2z^3\cos(xy^2z)-2x^3y^2z^4\sin(xy^2z)$

$\frac{\delta f(x,y,z)}{\delta y} = x^2z^3(\cos(xy^2z)-2xy^2z\sin(xy^2z))$

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Example Question #440 : Functions

The brightness of a lightbulb varies with its position on a board, and is given by the function

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

What is the affect of the variable x on the rate of change of the bulb's brightness?

Possible Answers:

$x^2y^2(2\sin(x+z)+x\cos(x+z))$

$2\sin(x+z)+x\cos(x+z)$

$yz(2\sin(x+z)+x\cos(x+z))$

$xy(2\sin(x+z)+x\cos(x+z))$

$xy^2(2\sin(x+z)+x\cos(x+z))$

Correct answer:

$xy(2\sin(x+z)+x\cos(x+z))$

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 and may represent large functions, and not just individual variables!

Our function is

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

and we're curious about the influence of x on the rate of its change, so we'll take the derivative with respect to x:

$\frac{\delta f(x,y,z)}{\delta x}=2xy\sin(x+z)+x^2y\cos(x+z)$

$\frac{\delta f(x,y,z)}{\delta x}=xy(2\sin(x+z)+x\cos(x+z))$

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Example Question #441 : Functions

What is the slope of the function f(x,y,z)=x^2yz^4 at coordinates (2,2,4) ?

Possible Answers:

$512\widehat{i}+1024\widehat{j}+2048\widehat{k}$

$2048\widehat{i}+2048\widehat{j}+2048\widehat{k}$

$2048\widehat{i}+1024\widehat{j}+256\widehat{k}$

$2048\widehat{i}+1024\widehat{j}+2048\widehat{k}$

$256\widehat{i}+512\widehat{j}+1024\widehat{k}$

Correct answer:

$2048\widehat{i}+1024\widehat{j}+2048\widehat{k}$

Explanation:

To solve this problem, it's worth introducing the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point.

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

For the function

f(x,y,z)=x^2yz^4 and coordinates (2,2,4)

Take the partial derivatives:

$\frac{\delta f}{\delta x}=2xyz^4$

$\frac{\delta f}{\delta x}=(2)(2)(2)(4)^4=2048$

$\frac{\delta f}{\delta y}=x^2z^4$

$\frac{\delta f}{\delta y}=(2)^2(4)^4=1024$

$\frac{\delta f}{\delta z}=4x^2yz^3$

$\frac{\delta f}{\delta z}=4(2)^2(2)(4)^3=2048$

The slope at this point is

$2048\widehat{i}+1024\widehat{j}+2048\widehat{k}$

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Example Question #442 : Functions

What is the slope of the function f(x,y,z)=x^2y+xz^3+y^2z^2 at point (5,1,3) ?

Possible Answers:

$41\widehat{i}+143\widehat{j}+237\widehat{k}$

$22\widehat{i}+34\widehat{j}+81\widehat{k}$

$37\widehat{i}+43\widehat{j}+141\widehat{k}$

$11\widehat{i}+21\widehat{j}+63\widehat{k}$

$73\widehat{i}+17\widehat{j}+141\widehat{k}$

Correct answer:

$37\widehat{i}+43\widehat{j}+141\widehat{k}$

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

The function we'll evaluate is

f(x,y,z)=x^2y+xz^3+y^2z^2

Take a derivative with respect to each variable and determine the value for the point (5,1,3)

$\frac{\delta f}{\delta x}=2xy+z^3$

$\frac{\delta f}{\delta x}=2(5)(1)+(3)^3=37$

$\frac{\delta f}{\delta y}=x^2+2yz^2$

$\frac{\delta f}{\delta y}=(5)^2+2(1)(3)^2=43$

$\frac{\delta f}{\delta z}=3xz^2+2y^2z$

$\frac{\delta f}{\delta z}=3(5)(3)^2+2(1)^2(3)=141$

The slope is thus

$37\widehat{i}+43\widehat{j}+141\widehat{k}$

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Example Question #443 : Functions

Find the slope of the function f(x,y,z)=xyz+x+y^2+z^3 at point (4,2,1)

Possible Answers:

$7\widehat{i}+9\widehat{j}+10\widehat{k}$

$3\widehat{i}+8\widehat{j}+11\widehat{k}$

$1\widehat{i}+1\widehat{j}+14\widehat{k}$

$5\widehat{i}+17\widehat{j}+2\widehat{k}$

$2\widehat{i}+6\widehat{j}+8\widehat{k}$

Correct answer:

$3\widehat{i}+8\widehat{j}+11\widehat{k}$

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

To restate the problem, we're looking for the slope of f(x,y,z)=xyz+x+y^2+z^3 at point (4,2,1)

$\frac{\delta f}{\delta x}=yz+1$ ; $\frac{\delta f}{\delta x}=(2)(1)+1=3$

$\frac{\delta f}{\delta y}=xz+2y$ ; $\frac{\delta f}{\delta y}=(4)(1)+2(2)=8$

$\frac{\delta f}{\delta z}=xy+3z^2$ ; $\frac{\delta f}{\delta z}=(4)(2)+3(1)^2=11$

Thus the slope is

$3\widehat{i}+8\widehat{j}+11\widehat{k}$

Report an Error

Example Question #444 : Functions

Find the slope of the function f(x,y,z)=sin(x)+2sin(y)+sin(2z) at point $(\pi,\pi,\pi)$

Possible Answers:

$1\widehat{i}-1\widehat{j}-1\widehat{k}$

$1\widehat{i}+2\widehat{j}+2\widehat{k}$

$-1\widehat{i}-2\widehat{j}+2\widehat{k}$

$1\widehat{i}+2\widehat{j}-2\widehat{k}$

$2\widehat{i}+2\widehat{j}+2\widehat{k}$

Correct answer:

$-1\widehat{i}-2\widehat{j}+2\widehat{k}$

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rule will be necessary:

Trigonometric derivative: $d[\sin(u)]=\cos(u)du$

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

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivative of $f(x,y,z)=\sin(x)+2\sin(y)+\sin(2z)$ at point $(\pi,\pi,\pi)$

$\frac{\delta f}{\delta x}=\cos(x);\frac{\delta f}{\delta x}=\cos(\pi)=-1$

$\frac{\delta f}{\delta y}=2\cos(y);\frac{\delta f}{\delta y}=2\cos(\pi)=-2$

$\frac{\delta f}{\delta z}=2\cos(2z);\frac{\delta f}{\delta z}=2\cos(2\pi)=2$

Thus the slope is $-1\widehat{i}-2\widehat{j}+2\widehat{k}$

Report an Error

Example Question #445 : Functions

Find the slope of the function $f(x,y)=e^{5x}+\cos(y)$ at the point $(\ln(2),\frac{\pi}{2})$

Possible Answers:

$160\widehat{i}+1\widehat{j}$

$80\widehat{i}-80\widehat{j}$

$160\widehat{i}-1\widehat{j}$

$32\widehat{i}+1\widehat{j}$

$32\widehat{i}-1\widehat{j}$

Correct answer:

$160\widehat{i}-1\widehat{j}$

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: d[e^u]=e^udu

Trigonometric derivative: $d[\cos(u)]=-\sin(u)du$

Product rule: d[uv]=udv+vdu

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

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y)=e^{5x}+\cos(y)$ at the point $(\ln(2),\frac{\pi}{2})$

$\frac{\delta f}{\delta x}=5e^{5x};\frac{\delta f}{\delta x}=5e^{5\ln(2)}=160$

$\frac{\delta f}{\delta y}=-\sin(y);\frac{\delta f}{\delta y}=-\sin(\frac{\pi}{2})=-1$

Thus the slope is $160\widehat{i}-1\widehat{j}$

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Example Question #446 : Functions

What is the slope of the function $f(x,y)=xe^{xy^2}$ at (2,3) ?

Possible Answers:

$18e^{18}\widehat{i}+12e^{18}\widehat{j}$

$6e^{18}\widehat{i}+18e^{18}\widehat{j}$

$19e^{18}\widehat{i}+12e^{18}\widehat{j}$

$1e^{18}\widehat{i}+12e^{18}\widehat{j}$

$3e^{18}\widehat{i}+3e^{18}\widehat{j}$

Correct answer:

$19e^{18}\widehat{i}+12e^{18}\widehat{j}$

Explanation:

To consider finding the slope, let's discuss the topic of the gradient.

For a function f(x_1,x_2,...,x_n) , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

$\frac{\delta f}{\delta x_1}\widehat{e_1}+\frac{\delta f}{\delta x_2}\widehat{e_2}+...+\frac{\delta f}{\delta x_n}\widehat{e_n}$

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: d[e^u]=e^udu

Product rule: d[uv]=udv+vdu

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

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

Take the partial derivatives of $f(x,y)=xe^{xy^2}$ at (2,3) :

$\frac{\delta f}{\delta x}=e^{xy^2}+xy^2e^{xy^2};\frac{\delta f}{\delta x}=e^{2(3)^2}+2(3)^2e^{2(3)^2}=e^{18}+18e^{18}=19e^{18}$

$\frac{\delta f}{\delta y}=2x^2ye^{xy^2};\frac{\delta f}{\delta y}=2(2)^2(3)e^{2(3)^2}=12e^{18}$

Thus the slope is

$19e^{18}\widehat{i}+12e^{18}\widehat{j}$

Report an Error

Example Question #1471 : Calculus

Find the limit of the following funciton.

$\lim_{x\rightarrow7}\frac{ x^2-49}{x-7}$

Possible Answers:

There is no limit.

Correct answer:

Explanation:

After plugging in for , we end up with $\frac{0}{0}$ .

This lets us know that there is still something to be done before coming up with the real solution.

In this case, we can see that $\frac{x^2-49\xtfrac{}{}}{x-7}=\frac{(x+7)(x-7))}{(x-7)}=(x+7)$ . Once we get here, we can see that pluggin in for will give us our answer, .

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Calculus 1 : How to find differential functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #438 : Functions

Example Question #251 : How To Find Differential Functions

Example Question #440 : Functions

Example Question #441 : Functions

Example Question #442 : Functions

Example Question #443 : Functions

Example Question #444 : Functions

Example Question #445 : Functions

Example Question #446 : Functions

Example Question #1471 : Calculus

All Calculus 1 Resources

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