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Calculus 1 : How to find differential functions

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Example Questions

Example Question #1031 : Differential Functions

Find the first derivative of the following function:

f(x)= 3x^2+2x+1

Possible Answers:

f'(x)= 6x+1

f'(x)= 6x+2

f'(x)= 6x+3

f'(x)= 5x+ 3

Correct answer:

f'(x)= 6x+2

Explanation:

To find the first derivative of this particular function is accomplished by applying the power rule which states,

$\frac{\mathrm{d} }{\mathrm{d} x}x^n= n(x)^{n-1}$

Applying the above rule to the equation,

f(x)= 3x^2+2x+1

results in,

$\\f'(x)= 2\cdot 3x^{2-1}+1\cdot 2x^{1-1} \\f'(x)=6x^1+2x^0$

f'(x)= 6x+2

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Example Question #851 : Other Differential Functions

Find the first derivative of the function $f(x)= 4x^2 + \sin(2x)$ .

Possible Answers:

$f'(x)= 8x-2\cos(2x)$

$f'(x)= 8x+ 2\cos(2x)$

$f'(x)= 8x-2\sin(2x)$

$f'(x)= 8x-\cos(2x)$

Correct answer:

$f'(x)= 8x+ 2\cos(2x)$

Explanation:

To take the derivative, you first use the power rule for differentiating,

$\frac{\mathrm{d} }{\mathrm{d} x}x^n= nx^{n-1}$ ,

then you use the chain rule,

f(g(x))'= f'(g(x))(g'(x)) .

Also recall the trigonometric rule for differentiating sine,

$(sin(u))'=cos(u)\frac{du}{dx}$

This produces,

$f'(x)= 2\cdot 4x^{2-1} + \cos(2x)\cdot 1\cdot2x^{1-1}$

$f'(x)= 8x +2\cos(2x)$ .

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Example Question #1031 : Differential Functions

Find the derivative.

$\frac{4x^2+3x}{x^3}$

Possible Answers:

$\frac{8x^4+3x^3-12x^2-9}{x^4}$

$\frac{-4x-6}{x^3}$

$\frac{8x^4-9}{x^4}$

$\frac{8x^4-9}{x^6}$

Correct answer:

$\frac{-4x-6}{x^3}$

Explanation:

Use the quotient rule to find the derivative which states,

$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ .

Given,

f(x)=4x^2+3x

g(x)=x^3

the derivatives can be found using the power rule which states,

$\frac{d}{dx}x^n=nx^{n-1}$

therefore,

$\\ f'(x)=2\cdot 4x^{2-1}+1\cdot 3x^{1-1}=8x+3 \\g'(x)=3x^{3-1}=3x^2$

Applying the quotient rule to our function we find the derivative to be as follows.

$\frac{d}{dx}\frac{4x^2+3x}{x^3}=\frac{x^3(8x+3)-(4x^2+3x)3x^2}{x^6}$

Simplify.

$\frac{8x^4+3x^3-12x^4-9x^3}{x^6}$

$\frac{-4x^4-6x^3}{x^6}$

$\frac{x^3(-4x-6)}{x^3(x^3)}=\frac{-4x-6}{x^3}$

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Example Question #851 : How To Find Differential Functions

Find the derivative.

-4x^3

Possible Answers:

-12x^2

12x^2

4x^3

-4x^2

Correct answer:

-12x^2

Explanation:

Use the power rule to find the derivative.

The power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$ .

Applying this rule to the function in the problem results in the following.

$\frac{d}{dx}(-4x^3)=3(-4)x^{3-1}=-12x^2$

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Example Question #852 : How To Find Differential Functions

Find the derivative.

4x-3

Possible Answers:

-4x

Correct answer:

Explanation:

Use the power rule to find the derivative.

The power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$ .

Applying this rule to each term of the function results in the following.

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

$\frac{d}{dx}-3=0$

Thus, the derivative is 4.

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Example Question #853 : Other Differential Functions

What is the equation for the slope of the tangent line to:

8x^2-4x+4

Possible Answers:

16x+4

16x-4

8x+4

8x-4

Correct answer:

16x-4

Explanation:

To find the equation for the slope of the tangent line, find the derivative.

To find the derivative, use the power rule.

The power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$ .

Applying the power rule to each term in the function results in,

$\frac{d}{dx}8x^2=16x$

$\frac{d}{dx}-4x=-4$

$\frac{d}{dx}4=0$ .

Thus, the derivative is 16x-4 .

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Example Question #853 : How To Find Differential Functions

Find the derivative when x=3 .

x^3+5x^2

Possible Answers:

Correct answer:

Explanation:

Use the power rule to find the derivative.

The power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$ .

Applying the power rule to each term within the function results in the following.

$\frac{d}{dx}x^3=3x^2$

$\frac{d}{dx}5x^2=10x$

Thus, the derivative is 3x^2+10x

Now, substitute for .

3(3^2)+10(3)=57

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Example Question #855 : Other Differential Functions

Find the derivative when x=2 .

5x^2-3x

Possible Answers:

Correct answer:

Explanation:

First, use the power rule to find the derivative.

The power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$ .

Applying the power rule to each term in the function results in the following.

$\frac{d}{dx}5x^2=10x$

$\frac{d}{dx}-3x=-3$

Thus, the derivative is 10x-3 .

Now, substitute 2 for x.

10(2)-3=17 .

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Example Question #856 : Other Differential Functions

Find the derivative.

$2x\cos (x)$

Possible Answers:

$2x\sin(x) -2\cos (x)$

$2x\sin(x) +2\cos (x)$

$-2x\sin(x) +2\cos (x)$

$-2x\sin(x) -2\cos (x)$

Correct answer:

$-2x\sin(x) +2\cos (x)$

Explanation:

Use the product rule to find the derivative.

The product rule states,

$\frac{d}{dx}f(x)g(x)=f(x)g'(x)+g(x)f'(x)$ .

Given,

f(x)=2x

g(x)=cos(x)

and recalling the trigonometry derivative for cosine is,

$\frac{d}{dx}cos(x)=-sin(x)$

the derivatives are as follows.

$f'(x)=1\cdot 2x^{1-1}=2$

g'(x)=-sin(x)

Therefore, using the product rule the derivative becomes,

$2x(-\sin x)+2\cos (x)$ .

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Example Question #857 : Other Differential Functions

Find the derivative.

2x^4-x^3

Possible Answers:

8x^3-x^2

8x^3+x^2

8x^3-3x^2

8x^3+3x^2

Correct answer:

8x^3-3x^2

Explanation:

Use the power rule to find the derivative.

The power rule states,

$\frac{d}{dx}x^n=nx^{n-1}$ .

Applying the power rule to each term in the function results in the following.

$\frac{d}{dx}2x^4=8x^3$

$\frac{d}{dx}(-x^3)=-3x^2$ .

Thus, the derivative is 8x^3-3x^2 .

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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 #1031 : Differential Functions

Example Question #851 : Other Differential Functions

Example Question #1031 : Differential Functions

Example Question #851 : How To Find Differential Functions

Example Question #852 : How To Find Differential Functions

Example Question #853 : Other Differential Functions

Example Question #853 : How To Find Differential Functions

Example Question #855 : Other Differential Functions

Example Question #856 : Other Differential Functions

Example Question #857 : Other Differential Functions

All Calculus 1 Resources

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