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Calculus 1 : Calculus

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Example Questions

Example Question #461 : Other Differential Functions

Find the derivative.

2x^6-3x^3+2x

Possible Answers:

9x^5-9x^2+2

12x^5-9x^2+2

12x-9x^2+2

12x^5-9x^2-2

Correct answer:

12x^5-9x^2+2

Explanation:

Use the power rule to find the derivative.

Remember the power rule is:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

$\frac{d}{dx}2x^6=12x^5$

$\frac{d}{dx}-3x^3=-9x^2$

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

Thus, the derivative is 12x^5-9x^2+2

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Example Question #462 : Other Differential Functions

Find the derivative at x=9 .

4x-5

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule

Recall the power rule:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

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

$\frac{d}{dx}-5=0$

Because the derivative is a constant , then the derivative at all points is .

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Example Question #463 : Other Differential Functions

Find the derivative at x=0 .

x^2+9x-3

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Recall the power rule:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

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

$\frac{d}{dx}9x=9$

Recall that the derivative of a constant is zero.

The derivative is 2x+9.

Now, substitute for .

2(0)+9

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Example Question #1681 : Calculus

Find the derivative.

$x^2\sin(x)$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Uses the product rule to find this derivative.

Recall the product rule:

$\\ \\ f(x)=(f(x)g(x))' \\ f'(x)=f'(x)g(x)+f(x)g'(x)$

$f'(x)=2x\sin (x)+x^2\cos (x)$

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Example Question #1682 : Calculus

Find the derivative.

$5x\cos (x)$

Possible Answers:

$-5x\cos (x)+5\sin (x)$

$5x\sin (x)+5\cos (x)$

$-5x\sin (x)-5\cos (x)$

$5\cos (x)-5x\sin (x)$

Correct answer:

$5\cos (x)-5x\sin (x)$

Explanation:

Use the product rule to find this derivative.

Recall the product rule:

$\\ \\ f(x)=(f(x)g(x))' \\ f'(x)=f'(x)g(x)+f(x)g'(x)$

$f'(x)=5\cos (x)-5x\sin (x)$

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Example Question #466 : Other Differential Functions

Find the slope of the tangent line at x=8 .

5x^2-7

Possible Answers:

Correct answer:

Explanation:

First, find the derivative using the power rule.

Recall the power rule:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

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

$\frac{d}{dx}-7=0$

The derivative is 10x .

Now, substitute for . The slope of the tangent line at x=8 is .

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Example Question #653 : Differential Functions

Find the derivative.

$-\frac{2}{x^4}$

Possible Answers:

$\frac{4}{x^5}$

$-\frac{4}{x^5}$

$-\frac{8}{x^5}$

$\frac{8}{x^5}$

Correct answer:

$\frac{8}{x^5}$

Explanation:

Use the quotient rule to find this derivative.

Recall the quotient rule:

$\\ \\ h(x)=\frac{f(x)}{g(x)} \\ \\ h'(x)=\frac{g(x)f'(x)-g'(x)f(x)}{(g(x))^2}$

$h'(x)=\frac{x^4(0)-2(4x^3)}{-x^8}$

$=\frac{-8x^3}{-x^8}=\frac{8x^3}{x^8}=\frac{8}{x^5}$

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Example Question #1688 : Calculus

Find the derivative at x=7 .

3x^2-2x^2+5x-4

Possible Answers:

Correct answer:

Explanation:

Begin by simplifying the expression.

x^2+5x-4

Now, find the derivative using the power rule.

Recall the power rule:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

2x+5

Finally, substitute for .

2(7)+5=19

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Example Question #1689 : Calculus

Find the slope of the tangent line at x=6 .

x^3

Possible Answers:

108

124

Correct answer:

108

Explanation:

First, find the derivative using the power rule.

Recall the power rule:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

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

Now, substitute for .

3(6^2)=3(36)=108

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Example Question #1683 : Calculus

Find the derivative.

5x^7

Possible Answers:

5x^6

35x^6

21x^5

6x^5

Correct answer:

35x^6

Explanation:

Use the power rule to find the derivative.

Recall the power rule:

$\\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}$

$\frac{d}{dx}5x^7=35x^6$

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #461 : Other Differential Functions

Example Question #462 : Other Differential Functions

Example Question #463 : Other Differential Functions

Example Question #1681 : Calculus

Example Question #1682 : Calculus

Example Question #466 : Other Differential Functions

Example Question #653 : Differential Functions

Example Question #1688 : Calculus

Example Question #1689 : Calculus

Example Question #1683 : Calculus

All Calculus 1 Resources

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