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Calculus 1 : Functions

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Example Questions

Example Question #291 : Differential Functions

Find the differential of the following equation.

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

Possible Answers:

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

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

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

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

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 #292 : Differential Functions

Find the differential of the following equation.

y=15x^2

Possible Answers:

dy=(15x)dx

dy=(30x^2)dx

dy=(30x)dx

dy=(30)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 #293 : Differential Functions

Find the differential of the following equation.

y=tan(x)+x^2

Possible Answers:

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

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

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

dy=(sec(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 #294 : 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(x)sin(x)+cos(x)tan(x))dx

dy=(sec^2(x)sin(x)+cos(x)sin(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 #295 : Differential Functions

Find the differential of the following equation.

y=ln(x)e^x

Possible Answers:

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

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

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

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

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 #296 : Differential Functions

Find the differential of the following equation.

y=x^2+e^x

Possible Answers:

dy=(2x+e^x)dx

dy=(2x-e^x)dx

dy=(2xe^x)dx

dy=(2+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 #297 : Differential Functions

Find the differential of the following equation.

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

Possible Answers:

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

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

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

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

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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Example Question #298 : Differential Functions

Find the differential of the following equation

y=cos(x)x^2

Possible Answers:

dy=(2xsin(x)+x^2sin(x))dx

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

dy=(2xcos(x)-x^2sin(x))dx

dy=(2cos(x)-x^2sin(x))dx

Correct answer:

dy=(2xcos(x)-x^2sin(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=(2xcos(x)+x^2sin(x))dx

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Example Question #299 : Differential Functions

Find the differential of the following equation.

$y=x^{3}e^{x}$

Possible Answers:

$dy=(3x^{2}e^{x}+x^{3}e^{x}})dx$

$dy=(3x^{2}e^{x}-x^{2}e^{x}})dx$

$dy=(3x^{2}e^{x}+x^{3}e^{x}})dx$

$dy=(3xe^{x}+x^{3}e^{x}})dx$

Correct answer:

$dy=(3x^{2}e^{x}+x^{3}e^{x}})dx$

Explanation:

The differential of is .

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

$dy=(3x^{2}e^{x}+x^{3}e^{x}})dx$

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Example Question #300 : Differential Functions

Find the differential of the following equation.

$y=4x^{3}-e^{-x}$

Possible Answers:

$dy=(12x^{2}+e^{-x})dx$

$dy=(12x+e^{-x})dx$

$dy=(12x^{2}+e^{x})dx$

$dy=(12x^{2}-e^{-x})dx$

Correct answer:

$dy=(12x^{2}+e^{-x})dx$

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:

$dy=(12x^{2}+e^{-x})dx$

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Calculus 1 : Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #291 : Differential Functions

Example Question #292 : Differential Functions

Example Question #293 : Differential Functions

Example Question #294 : Differential Functions

Example Question #295 : Differential Functions

Example Question #296 : Differential Functions

Example Question #297 : Differential Functions

Example Question #298 : Differential Functions

Example Question #299 : Differential Functions

Example Question #300 : Differential Functions

All Calculus 1 Resources

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