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Calculus 1 : How to find differential functions

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Example Questions

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 #309 : Differential Functions

Calculate the differential for the following function.

y=2+3x-x^2

Possible Answers:

dy=(3-2x)

dy=(3x-2x)dx

dy=(-2x)dx

dy=(-3-2x)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 #121 : Other Differential Functions

Compute the following differential.

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

Possible Answers:

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

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

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

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

dy=(12x+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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Example Question #122 : Other Differential Functions

Calculate the differential for the following function.

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

Possible Answers:

dy=x^2dx

dy=(4x)dx

$dy=\frac{2x}{x^2}dx$

dy=(2x)dx

dy=xdx

Correct answer:

dy=xdx

Explanation:

To solve this problem, you may use the quotient rule for finding derivatives. The quotient rule stipulates that for a function

$y=\frac{a}{b}$ , $y'=\frac{ba'-ab'}{b^2}$ .

In this problem, a=x^2 and b=2 .

Thus,

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

$\frac{dy}{dx}=\frac{4x}{4}$

$\frac{dy}{dx}=x$ .

Therefore,

$\frac{dy}{dx}=x$ and dy=xdx .

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Example Question #123 : Other Differential Functions

Calculate the differential of the following function.

$y=\frac{1}{2}g^2+\frac{3}{5}g^3+\frac{2}{3}g$

Possible Answers:

$dy=\left(g^2+\frac{9}{5}g^2+\frac{2}{3}\right)dg$

$dy=\left(g+\frac{9}{5}g^2\right)dg$

$dy=\left(2g+\frac{9}{5}g^2+\frac{2}{3}\right)dg$

$dy=\left(g+\frac{9}{5}g^2+\frac{2}{5}\right)dg$

$dy=\left(g+\frac{9}{5}g^2+\frac{2}{3}\right)dg$

Correct answer:

$dy=\left(g+\frac{9}{5}g^2+\frac{2}{3}\right)dg$

Explanation:

Using the power rule, we can find the derivative of each part of the function. When using the power rule you multiply the coefficient by the exponent then decrease the exponent by one.

The derivative of $\frac{1}{2}g^2$ is $\frac{1}{2}\cdot 2g^{2-1}=g$ .

The derivative of $\frac{3}{5}g^3$ is $\frac{9}{5}g^2$ .

The derivative of $\frac{2}{3}g$ is $\frac{2}{3}$ .

When these derivatives are added together,

$\frac{dy}{dg}=g+\frac{9}{5}g^2+\frac{2}{3}$ .

Thus,

$dy=\left(g+\frac{9}{5}g^2+\frac{2}{3}\right)dg$

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Example Question #124 : Other Differential Functions

Calculate the differential for the following function.

$y=\frac{2x}{x-1}$

Possible Answers:

$dy=({(x-1)^2})dx$

$dy=\left(-\frac{2x}{(x-1)^2}\right)dx$

$dy=\left(\frac{2}{(x-1)^2}\right)dx$

$dy=\left(-\frac{2}{(x-1)^2}\right)dx$

$dy=2x-1 \ dx$

Correct answer:

$dy=\left(-\frac{2}{(x-1)^2}\right)dx$

Explanation:

Use the quotient rule to find this answer. The quotient rule dictates that for a function

$y=\frac{a}{b}$ , $y'=\frac{b(a')-a(b')}{b^2}$ .

For this particular question, a=2x and b=x-1 .

Apply the quotient rule to this function:

$\frac{dy}{dx}=\frac{2x-2-2x}{(x-1)^2}=-\frac{2}{(x-1)^2}$

$dy=\left(-\frac{2}{(x-1)^2}\right)dx$

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Example Question #125 : Other Differential Functions

Calculate the differential for the following function.

$y=\sqrt{x}$

Possible Answers:

$dy=x^{\frac{1}{2}} dx$

$dy=4\sqrt{x} dx$

$dy=\left(\frac{1}{2\sqrt{x}}\right)dx$

$dy=\left(\frac{1}{\sqrt{x}}\right)dx$

$dy=\left(\frac{2}{\sqrt{x}}\right)dx$

Correct answer:

$dy=\left(\frac{1}{2\sqrt{x}}\right)dx$

Explanation:

For finding the derivative of a root, it is helpful to turn the root into a power.

For example, in this problem it is helpful to turn the $\sqrt{x}$ into $x^\frac{1}{2}$ .

Now, we can easily apply the power rule,

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

which yields the answer

$dy=\left(\frac{1}{2\sqrt{x}}\right)dx$ .

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Example Question #311 : Differential Functions

Calculate the differential for the following function.

$y=12x^{2}-\sqrt{x}$

Possible Answers:

$dy=\left(24x-\frac{1}{\sqrt{x}}\right)dx$

$dy=\left(\frac{1}{2\sqrt{x}}\right)dx$

$dy=\left(24x^2-\frac{1}{2\sqrt{x}}\right)dx$

dy=24x+12x^2 dx

$dy=\left(24x-\frac{1}{2\sqrt{x}}\right)dx$

Correct answer:

$dy=\left(24x-\frac{1}{2\sqrt{x}}\right)dx$

Explanation:

Using the power rule, we can solve this problem. The power rule states to multiply the coefficient with the exponent of the term then decrease the exponent by one.

The derivative of 12x^2 is 24x .

The derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$ .

Thus,

$\frac{dy}{dx}=24x-\frac{1}{2\sqrt{x}}$ and

$dy=\left(24x-\frac{1}{2\sqrt{x}}\right)dx$

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Example Question #126 : Other Differential Functions

Calculate the differential for the following.

y=x^2+3x+4x^3

Possible Answers:

$dy=(4x^3+2)\ dx$

dy=(2x^2+3+12x^2)dx

dy=(2x+3+12x^2)dx

dy=(2x+3x+12x^2)dx

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

Correct answer:

dy=(2x+3+12x^2)dx

Explanation:

Use the power rule to differentiate this function. The power rule states to multiply the coefficient by the exponent then decrease the exponent by one.

The derivative of x^2 is .

The derivative of is .

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

Thus,

$\frac{dy}{dx}=2x+3+12x^2$ and dy=(2x+3+12x^2)dx

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Example Question #1341 : Calculus

Calculate the differential for the following.

$y=\frac{6x^2}{2-x}$

Possible Answers:

$dy=\frac{24x-6x^2+3}{(2-x)^2}\ dx$

$dy=\frac{24x-6x^2}{(2-x)^2}\ dx$

$dy=\frac{24x-6x^2}{4-x^2}\ dx$

$dy=\frac{4x^3}{2-x^2}\ dx$

$dy=\frac{24x^2-6x^2}{(2-x)^2}\ dx$

Correct answer:

$dy=\frac{24x-6x^2}{(2-x)^2}\ dx$

Explanation:

Use the quotient rule to find the solution to this problem. The quotient rule stipulates that for a function

$y=\frac{a}{b}$ , $y'=\frac{b(a')-a(b')}{b^2}$

In this problem, a=6x^2 and b=2-x .

Apply the quotient rule:

$\frac{dy}{dx}=\frac{2-x(12x)-6x^2(-1)}{(2-x)^2}$

$\frac{dy}{dx}=\frac{24x-6x^2}{(2-x)^2}$

$dy=\left(\frac{24x-6x^2}{(2-x)^2}\right)dx$

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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 #302 : Differential Functions

Example Question #309 : Differential Functions

Example Question #121 : Other Differential Functions

Example Question #122 : Other Differential Functions

Example Question #123 : Other Differential Functions

Example Question #124 : Other Differential Functions

Example Question #125 : Other Differential Functions

Example Question #311 : Differential Functions

Example Question #126 : Other Differential Functions

Example Question #1341 : Calculus

All Calculus 1 Resources

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