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Calculus 1 : How to find differential functions

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Example Questions

Example Question #1491 : Calculus

Find the slope of the tangent line at x=11 .

$x^{3}(x+3)$

Possible Answers:

1893

1794

1894

6413

Correct answer:

6413

Explanation:

Begin by finding the derivative using the product rule.

The product rule is,

f(x)g(x)'=f(x)g'(x)+g(x)f'(x) .

Applying this rule to our function where

$f(x)=x^3\rightarrow f'(x)=3x^2$

$g(x)=x+3\rightarrow g'(x)=1$

we get,

x^3(1)+(x+3)3x^2

x^3+3x^3+9x^2

4x^3+9x^2

Now, substitute 11 for x.

$4(11)^{3}+9\cdot11^{2}$

=5324+1089

=6413

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Example Question #461 : Functions

Find the slope of the tangent line at x=1 .

$(x+3)x^{4}$

Possible Answers:

Correct answer:

Explanation:

Begin by finding the derivative using the product rule.

The product rule is,

f(x)g(x)'=f(x)g'(x)+g(x)f'(x) .

Applying this rule to our function where

$f(x)=(x+3)\rightarrow f'(x)=1$

$g(x)=x^4\rightarrow g'(x)=4x^3$

we get,

$(x+3)4x^3+x^{4}(1)$

$4x^{4}+12x^{3}+x^{4}$

$5x^{4}+12x^{3}$

Now, substitute 1 for x.

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Example Question #461 : Functions

What is the slope of the tanget line at x=2.5 ?

$2x^{4}-3x^{2}+2$

Possible Answers:

110

115

105

100

Correct answer:

110

Explanation:

First, find the derivative using the power rule which states, $(x^n)'=nx^{n-1}$ .

Applying the power rule to each term in the function we get,

$8x^{3}-6x$

Now, plug in 2.5 for x.

$8(2.5^{3})-6(2.5)$

125-15=110

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Example Question #281 : Other Differential Functions

Find the slope of the tangent line at x=1.5 .

$\frac{x^{2}}{4}$

Possible Answers:

$\frac{1}{2}$

$\frac{3}{4}$

$\frac{1}{4}$

Correct answer:

$\frac{3}{4}$

Explanation:

First, find the derivative using the quotient rule which states,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

In this particular case,

$f(x)=x^2\rightarrow f'(x)=2x$

$g(x)=4\rightarrow g'(x)=0$ .

Therefore the derivative becomes,

$\frac{4(2x)-x^2(0)}{4^2}=\frac{4(2x)}{16}$

$=\frac{8x}{16}=\frac{x}{2}$

Now, substitute 1.5 for x.

$\frac{1.5}{2}=\frac{3}{4}$

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Example Question #281 : Other Differential Functions

Find the derivative.

$\frac{4}{2x}$

Possible Answers:

$-\frac{2}{x^{2}}$

$-x^{2}$

$-\frac{2}{x^{3}}$

Correct answer:

$-\frac{2}{x^{2}}$

Explanation:

Use the quotient rule to find the derivative which states,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

In this particular case,

$f(x)=4\rightarrow f'(x)=0$

$g(x)=2x\rightarrow g'(x)=2$ .

Therefore the derivative becomes,

$\frac{2x(0)-4(2)}{(2x)^2}=\frac{-8}{4x^{2}}$

$=-\frac{2}{x^{2}}$

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Example Question #1501 : Calculus

Find the derivative.

$\frac{2x}{x^{3}}$

Possible Answers:

$\frac{-4}{x^{4}}$

$\frac{-4}{x^3}$

$\frac{-3}{x^{3}}$

$\frac{-2}{x^5}$

Correct answer:

$\frac{-4}{x^3}$

Explanation:

Use the quotient rule to find the derivative which states,

$\frac{f(x)}{g(x)}'=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

In this particular case,

$f(x)=2x\rightarrow f'(x)=2$

$g(x)=x^3\rightarrow g'(x)=3x^2$ .

Therefore, the derivative becomes,

$=\frac{2x^{3}-2x(3x^{2})}{x^{6}}$

$=\frac{-4x^{3}}{x^6}$

$=\frac{-4}{x^3}$

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Example Question #287 : Other Differential Functions

What is the slope of the function f(x,y)=x^y at (e,4) ?

Possible Answers:

$(4e^4,e^4\ln(4))$

(4e^3,e^4)

$(4e^3,e^4\ln(4))$

(e^3,e^4)

(e^4,e^4)

Correct answer:

(4e^3,e^4)

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:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Note that 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)=x^y at (e,4)

$\frac{\delta f}{\delta x}=yx^{y-1}=4e^3$

$\frac{\delta f}{\delta y}=x^y\ln(x)=e^4\ln(e)=e^4$

The slope is

(4e^3,e^4)

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Example Question #288 : Other Differential Functions

Find the slope of the function f(x,y)=x^yy^x at the point (2,4)

Possible Answers:

(305.4,866.9)

(457.2,203.3)

(203.3,457.2)

(866.9,305.4)

Correct answer:

(866.9,305.4)

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: $d[a^u]=a^u\ln(a)du$

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.

Find the partial derivatives of f(x,y)=x^yy^x at the point (2,4)

$\frac{\delta f}{\delta x}=x^{y-1}y^{x+1}+x^{y}y^{x}ln(y)=2^34^3+2^44^2ln(4)=866.9$

$\frac{\delta f}{\delta y}=x^{y}y^{x}ln(x)+x^{y+1}y^{x-1}=2^44^2ln(2)+2^54=305.4$

The slope is

(866.9,305.4)

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Example Question #289 : Other Differential Functions

Find the slope of the function $f(x,y)=x^{2+2y}$ at (1,7) .

Possible Answers:

(2,2)

(0,16)

(4,8)

(8,4)

(16,0)

Correct answer:

(16,0)

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: $d[a^u]=a^u\ln(a)du$

Note that 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 $f(x,y)=x^{2+2y}$ at (1,7)

$\frac{\delta f}{\delta x}=(2+2y)x^{1+2y}=(2+14)1^{15}=16$

$\frac{\delta f}{\delta y}=2x^{2+2y}\ln(x)=2(1^{16})\ln(1)=0$

The slope is

(16,0)

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Example Question #290 : Other Differential Functions

Find the slope of the function $f(x,y,z)=x^{yz^2}$ at the point (2,3,1)

Possible Answers:

(5.5,33.3,12)

(12,5.5,33.3)

(33.3,5.5,12)

(12,33.3,5.5)

(5.5,12,33.3)

Correct answer:

(12,5.5,33.3)

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:

Derivative of an exponential: $d[a^u]=a^u\ln(a)du$

Note that 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 $f(x,y,z)=x^{yz^2}$ at the point (2,3,1)

$\frac{\delta f}{\delta x}=yz^2x^{yz^2-1}=3(1^2)2^{3(1^2)-1}=12$

$\frac{\delta f}{\delta y}=z^2x^{yz^2}\ln(x)=1^2(2^{3(1^2)})\ln(2)=5.5$

$\frac{\delta f}{\delta z}=2yzx^{yz^2}\ln(x)=2(3)(1)(2^{3(1^2)})\ln(2)=33.3$

The slope is

(12,5.5,33.3)

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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 #1491 : Calculus

Example Question #461 : Functions

Example Question #461 : Functions

Example Question #281 : Other Differential Functions

Example Question #281 : Other Differential Functions

Example Question #1501 : Calculus

Example Question #287 : Other Differential Functions

Example Question #288 : Other Differential Functions

Example Question #289 : Other Differential Functions

Example Question #290 : Other Differential Functions

All Calculus 1 Resources

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