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Calculus 1 : Differential Functions

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Example Questions

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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Example Question #302 : Differential Functions

Compute the differential for the following.

y=sin(x)cos(x)

Possible Answers:

dy=[-sin^2(x)+cos^2(x)]dx

dy=[sin^2(x)+cos^2(x)]dx

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

dy=[-sin(x)+cos^2(x)]dx

dy=cos^2(x)dx

Correct answer:

dy=[-sin^2(x)+cos^2(x)]dx

Explanation:

To solve this problem, you must use the product rule of finding derivatives.

For any function y=ab , y'=a(b')+b(a') .

In this problem, the product rule yields

a=sin(x), a'=cos(x)

b=cos(x), b'=-sin(x)

$\frac{dy}{dx}=sin(x)\cdot(-sin(x))+cos(x)\cdot cos(x)$

dy=[-sin^2(x)+cos^2(x)]dx .

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Example Question #121 : How To Find Differential Functions

Calculate the differential for the following function.

y=2+3x-x^2

Possible Answers:

dy=(-3-2x)dx

dy=(3-2x)

dy=(3x-2x)dx

dy=(-2x)dx

dy=(3-2x)dx

Correct answer:

dy=(3-2x)dx

Explanation:

This differential can be found by utilizing the power rule,

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

The original equation is y=2+3x-x^2 .

Using the power rule on each term we see that the derivative of is . The derivative of a constant is always zero.

The dervative of is

$1\cdot3 x^{1-1}=3x^0=3$ .

The derivative of -x^2 is

$-1\cdot 2x^{2-1}=-2x^1=-2x$ .

Thus,

$\frac{dy}{dx}=3-2x$

Multiply to the right side to get the final solution.

dy=(3-2x)dx

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Example Question #122 : How To Find Differential Functions

Compute the following differential.

$y=4x^3+8x^2+32x+\pi$

Possible Answers:

$dy=(12x^2+16x+32+\Pi )dx$

dy=(12x+16x+32)dx

dy=(12x^2-16x+32)dx

dy=(12x^2+16x+32)dx

dy=(12x^2-16x-32)dx

Correct answer:

dy=(12x^2+16x+32)dx

Explanation:

Using the power rule, we can find the derivative of each part of the function. The power rule states to multiply the coefficient of the term by the exponent then decrease the exponent by one.

The derivative of 4x^3 is 12x^2 .

The derivative of 8x^2 is 16x .

The derivative of 32x is .

And finally, because $\pi$ is a constant, the derivative of $\pi$ is .

Thus, when we add the parts together, the derivative is

$\frac{dy}{dx}=12x^2+16x+32$

and

dy=(12x^2+16x+32)dx

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Calculus 1 : Differential Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

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

Example Question #302 : Differential Functions

Example Question #121 : How To Find Differential Functions

Example Question #122 : How To Find Differential Functions

All Calculus 1 Resources

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