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Calculus 1 : Other Differential Functions

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Example Questions

Example Question #291 : Functions

Find the differential of the following equation

y=cos(x)x^2

Possible Answers:

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

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

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

dy=(2xcos(x)+x^2)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 #292 : Functions

Find the differential of the following equation.

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

Possible Answers:

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

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

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

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

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 #113 : How To Find Differential Functions

Find the differential of the following equation.

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

Possible Answers:

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

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

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

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

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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Example Question #114 : How To Find Differential Functions

$f(x)=(3x^{2}+2x)^{3}$

Find

$\frac{d}{dx}f(x)$ .

Possible Answers:

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

(18x+6)(3x^2+2x)^2

(3x^2+2x)^3(6x+2)

(6x+2)^3

3(6x+2)^2

Correct answer:

(18x+6)(3x^2+2x)^2

Explanation:

Let g(x)=3x^2+2x .

Then f(x) = g(x)^3 .

By the chain rule,

$\frac{df}{dx}=\frac{df}{dg}\frac{dg}{dx}$

$\frac{df}{dg}=3g(x)^2$ , $\frac{dg}{dx}=6x+2$

Plugging everything in we get

$\frac{df}{dx}=3(3x^2+2x)^2(6x+2)=(18x+6)(3x^2+2x)^2$

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Example Question #301 : Differential Functions

Let $f(x)=\frac{x^2}{e^x+3x^3}$

Find

$\frac{d}{dx}f(x)$ .

Possible Answers:

$\frac{2x(e^x+3x^3)-x^2(e^x+9x^2)}{(e^x+3x^3)^2}$

$\frac{(e^x+3x^3)-x^2(e^x+9x^2)}{(e^x+3x^3)}$

$\frac{(e^x+3x^3)^2-x^2(e^x+9x^2)}{(e^x+3x^3)^2}$

$\frac{(e^x+3x^3)-x^2(e^x+9x^2)}{(e^x+3x^3)^2}$

$\frac{2x(e^x+3x^3)-x^2(e^x+9x^2)}{(e^x+3x^3)}$

Correct answer:

$\frac{2x(e^x+3x^3)-x^2(e^x+9x^2)}{(e^x+3x^3)^2}$

Explanation:

Let g(x)=x^2 and $h(x)=(e^x+ex^3)^{-1}$ .

So f(x)=g(x)h(x) .

By the product rule:

$\frac{df}{dx}=g'h+h'g$

Where $h'=\frac{dh}{dx}$ and $g'=\frac{dg}{dx}$ .

Therefore,

g'=2x

$h'=-(e^x+3x^3)^{-2}(e^x+9x^2)$

Plugging everything in and simplifying we get:

$\frac{df}{dx}=\frac{2x(e^x+3x^3)-x^2(e^x+9x^2)}{(e^x+3x^3)^2}$

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Example Question #111 : Other Differential Functions

Let f(x)=ln[x^3e^x]

Find

$\frac{d}{dx}f(x)$ .

Possible Answers:

ln[3x^2e^x+x^3e^x]

ln[x^3e^x]

$\frac{1}{x^3e^x}$

$\frac{3}{x}+1$

$ln[x^3e^x]^{-1}(3x^2e^x+x^3e^x)$

Correct answer:

$\frac{3}{x}+1$

Explanation:

We can simplify the function by using the properties of logarithms.

f(x)=ln[x^3]+ln[e^x]=3ln[x]+x

With the simplified form, we can now find the derivative using the power rule which states,

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$ .

Also we will need to use the product rule which is,

$f(x)=g(x)h(x)\rightarrow f'(x)=g(x)h'(x)+h(x)g'(x)$ .

Remember that the derivative of $ln(x)\rightarrow \frac{1}{x}$ .

Applying these rules we find the derivative to be as follows.

$\frac{df}{dx}=\frac{d}{dx}[3ln[x]+x]=\frac{3}{x}+1$

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Example Question #304 : Differential Functions

Let f(x)=3^x .

Find

$\frac{d}{dx}f(x)$ .

Possible Answers:

ln(x)3^x

ln(3)3^x

log_3(3)3^x

$(x-1)3^{x-1}$

$x3^{x-1}$

Correct answer:

ln(3)3^x

Explanation:

For a function of the form f(x)=a^x the derivative is by definition:

$\frac{d}{dx}f(x)=f'(x)=ln(a)a^x$ .

Therefore,

$f(x)=3^x\rightarrow f'(x)=3^xln(x)$ .

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Example Question #305 : Differential Functions

Let f(x)=sin^2(x)

Find

$\frac{d}{dx}f(x)$ .

Possible Answers:

cos^2(x)

2sin(x)

2cos(x)sin(x)

cos(x)+sin(x)

sin^2(x)cos^2(x)

Correct answer:

2cos(x)sin(x)

Explanation:

Recall that,

sin^2(x)=sin(x)sin(x)

Using the product rule

$\frac{d}{dx}sin^2(x)=cos(x)sin(x)+sin(x)cos(x)=2cos(x)sin(x)$

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Example Question #301 : Differential Functions

Compute the differential for the following.

y=9t^2+24t-50

Possible Answers:

dy=(18t^2+24)dt

dy=(18t+24t)dt

dy=(18t+24)^2dt

dy=(18t+24)dt

dy=(18t^4+24t^2-50t)dt

Correct answer:

dy=(18t+24)dt

Explanation:

To compute the differential of the function we will need to use the power rule which states,

$y=x^n \rightarrow dy=nx^{n-1}$ .

Applying the power rule we get:

$\frac{dy}{dt}=18t+24$

From here solve for dy:

dy=(18t+24)dt

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Example Question #1331 : Calculus

Compute the differential for the following function.

y=7x^3+cos(x)

Possible Answers:

dy=(21x^2+sin(x))dx

dy=(21x^2-sin(x)cos(x))dx

dy=(21x^2-cos(x))dx

dy=(21x^2-sin(x))dx

dy=(21x^2+cos(x))dx

Correct answer:

dy=(21x^2-sin(x))dx

Explanation:

Using the power rule,

$y=x^n \rightarrow dy=nx^{n-1}dx$

the derivative of 7x^3 becomes 21x^2 .

Using trigonometric identities, the derivative of cos(x) is -sin(x) .

Therefore,

$\frac{dy}{dx}=21x^2-sin(x)$

dy=(21x^2-sin(x))dx

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Calculus 1 : Other Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #291 : Functions

Example Question #292 : Functions

Example Question #113 : How To Find Differential Functions

Example Question #114 : How To Find Differential Functions

Example Question #301 : Differential Functions

Example Question #111 : Other Differential Functions

Example Question #304 : Differential Functions

Example Question #305 : Differential Functions

Example Question #301 : Differential Functions

Example Question #1331 : Calculus

All Calculus 1 Resources

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