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Calculus 1 : How to find differential functions

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Example Questions

Example Question #101 : How To Find Differential Functions

Differentiate the trigonometric function.

$\cos(3x)$

Possible Answers:

$-3\sin(x)$

$-\cos(3x)$

$3\sin(3x)$

$-3\cos(3x)$

$\sin(3x)$

Correct answer:

$-3\sin(x)$

Explanation:

We can use the chain rule to differentiate, which states we will need to multiply the derivative of the outside function by the derivative of the inside function. We find the derivative of the inside function, , to be . The derivative of the outside function $\cos(x)$ , will be $-\sin(x)$ . Multiplying these values together results in $-3\sin(3x)$ .

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Example Question #101 : How To Find Differential Functions

Find the differential of the following equation.

y=x^3+2x

Possible Answers:

dy=(3x^2)dx

dy=3x+2 dx

y=dx(6)

dy=(3x^2+2)dx

Correct answer:

dy=(3x^2+2)dx

Explanation:

To find the differential, take the derivative of each term as follows.

The derivative of anything in the form of x^n is $n{x^{n-1}}$ so applying that rule to all of the terms yields:

dy=(3x^2+2)dx

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Example Question #103 : Other Differential Functions

Find the differential of the following equation.

y=sinx+x^2

Possible Answers:

dy(sinx +2x)dx

dy=(cosx +2x)dx

dy=(cosxx^2)dx

dy=(cosx)dx

Correct answer:

dy=(cosx +2x)dx

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of sin(x) is cos(x) and anything in the form of x^n is $n{x^{n-1}}$ , so applying that rule to all of the terms yields:

dy=(cosx +2x)dx

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Example Question #104 : Other Differential Functions

Find the differential of the following equation.

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

Possible Answers:

$dy=\frac{{cos(x)-(x^2)(sinx)}}{{cos^2(x)}}dx$

$dy=\frac{{sin(x)(2x)-(x^2)(-sinx)}}{{sin^2(x)}}dx$

$dy=\frac{{cos(x)(2x)-(x^2)(-sinx)}}{{cos^2(x)}}dx$

$dy=\frac{{cos(x)(2x)-(x^2)(-sinx)}}{{sin^2(x)}}dx$

Correct answer:

$dy=\frac{{cos(x)(2x)-(x^2)(-sinx)}}{{cos^2(x)}}dx$

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as you apply the quotient rule.

The quotient rule is:

$\frac{(denominator)(d(numerator))-(numerator)(d(denominator))}{denominator^2}$ ,

so applying that rule to the equation yields:

$dy=\frac{{cos(x)(2x)-(x^2)(-sinx)}}{{cos^2(x)}}dx$

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Example Question #105 : Other Differential Functions

Find the differential of the following equation.

y=15x^2

Possible Answers:

dy=(30x)dx

dy=(15x)dx

dy=(30)dx

dy=(30x^2)dx

Correct answer:

dy=(30x)dx

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative as follows.

The derivative of anything in the form of x^n is $n{x^{n-1}}$ , so applying that rule to all of the terms yields:

dy=(30x)dx

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Example Question #106 : Other Differential Functions

Find the differential of the following equation.

y=tan(x)+x^2

Possible Answers:

dy=(sec^2(x)+2)dx

dy=(sec(x)+2x)dx

dy=(sec^2(x)*2x)dx

dy=(sec^2(x)+2x)dx

Correct answer:

dy=(sec^2(x)+2x)dx

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of tan(x) is sec^2(x) , and derivative of anything in the form of x^n is $n{x^{n-1}}$ , so applying that rule to all of the terms yields:

dy=(sec^2(x)+2x)dx

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Example Question #107 : Other Differential Functions

Find the differential of the following equation.

$y=sin(x)\cdot tan(x)$

Possible Answers:

dy=(sec^2(x)sin(x)+sin(x)tan(x))dx

dy=(sec^2(x)sin(x)+cos(x)sin(x))dx

dy=(sec(x)sin(x)+cos(x)tan(x))dx

dy=(sec^2(x)sin(x)+cos(x)tan(x))dx

Correct answer:

dy=(sec^2(x)sin(x)+cos(x)tan(x))dx

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as you apply the product rule.

The product rule is

$(second\quad term)\cdot d(first\quad term) + (first\quad term)\cdot d(second\quad term)$ , so applying that rule to the equation yields:

dy=(sec^2(x)sin(x)+cos(x)tan(x))dx

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Example Question #1321 : Calculus

Find the differential of the following equation.

y=ln(x)e^x

Possible Answers:

dy=(e^x+ln(x)e^x)dx

$dy=\left(\frac{e^x}{x}+e^x\right)dx$

$dy=\left(\frac{e^x}{x}-ln(x)e^x\right)dx$

$dy=\left(\frac{e^x}{x}+ln(x)e^x\right)dx$

Correct answer:

$dy=\left(\frac{e^x}{x}+ln(x)e^x\right)dx$

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as you apply the product rule.

The product rule is:

$(second\quad term)\cdot d(first\quad term) + (first\quad term)\cdot d(second\quad term)$ , so applying that rule to the equation yields:

$dy=\left(\frac{e^x}{x}+ln(x)e^x\right)dx$

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Example Question #1325 : Calculus

Find the differential of the following equation.

y=x^2+e^x

Possible Answers:

dy=(2+e^x)dx

dy=(2xe^x)dx

dy=(2x-e^x)dx

dy=(2x+e^x)dx

Correct answer:

dy=(2x+e^x)dx

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of anything in the form of x^n is $n{x^{n-1}}$ , and the derivative of e^x is e^x so applying that rule to all of the terms yields [correct answer]:

dy=(2x+e^x)dx

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Example Question #108 : Other Differential Functions

Find the differential of the following equation.

$y=e^{-x}+sin(x)$

Possible Answers:

$dy=(e^{-x}+cos(x))dx$

$dy=(e^{-x}-sin(x))dx$

$dy=(-e^{-x}-cos(x))dx$

$dy=(-e^{-x}+cos(x))dx$

Correct answer:

$dy=(-e^{-x}+cos(x))dx$

Explanation:

The differential of is .

To find the differential of the right side of the equation, take the derivative of each term as follows.

The derivative of $e^{-x}$ is $-e^{-x}$ , and the derivative of sin(x) is cos(x) , so applying that rule to all of the terms yields [correct answer]:

$dy=(-e^{-x}+cos(x))dx$

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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 #101 : How To Find Differential Functions

Example Question #101 : How To Find Differential Functions

Example Question #103 : Other Differential Functions

Example Question #104 : Other Differential Functions

Example Question #105 : Other Differential Functions

Example Question #106 : Other Differential Functions

Example Question #107 : Other Differential Functions

Example Question #1321 : Calculus

Example Question #1325 : Calculus

Example Question #108 : Other Differential Functions

All Calculus 1 Resources

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