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Calculus 1 : How to find differential functions

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Example Questions

Example Question #931 : Functions

Let f(x)=x^3+3 on the interval (9,10) . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.

Possible Answers:

9.008

135.5

28.5

9.504

Correct answer:

9.504

Explanation:

The mean value theorem states that for a planar arc passing through a starting and endpoint (a,b);a<b , there exists at a minimum one point, , within the interval (a,b) for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.

Meanvaluetheorem

In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.

$f'(c)=\frac{f(b)-f(a)}{b-a}=\frac{Rise}{Run};a<c<b$

Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.

First, find the two function values of f(x)=x^3+3 on the interval (9,10)

f(9)=9^3+3=732

f(10)=10^3+3=1003

Then take the difference of the two and divide by the interval.

$\frac{1003-732}{10-9}=271$

Now find the derivative of the function; this will be solved for the value(s) found above.

f'(x)=3x^2

3x^2=271

Using a calculator, we find the solution x=9.504 , which fits within the interval (9,10) , satisfying the mean value theorem.

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Example Question #751 : How To Find Differential Functions

Find the derivative.

$\frac{1}{x^2}$

Possible Answers:

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

$\frac{-2x}{x^3}$

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

$\frac{-2}{x^3}$

Correct answer:

$\frac{-2}{x^3}$

Explanation:

Use the quotient rule to find the derivative.

$\frac{(x^2)(0)-(1)(2x)}{(x^2)^2}=\frac{-2x}{x^4}=\frac{-2}{x^3}$

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Example Question #752 : How To Find Differential Functions

Find the derivative.

2x-8

Possible Answers:

Correct answer:

Explanation:

Use the power rule to calculate the derivative.

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

Recall that the derivative of a constant is zero.

Thus, the derivative is 2.

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Example Question #753 : How To Find Differential Functions

Find the derivative.

y^3+2y+y^2

Possible Answers:

3y^2+2y+2

2y^2

y^2+2

2y^2+2y

Correct answer:

3y^2+2y+2

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dy}y^3=3y^2$

$\frac{d}{dy}y^2=2y$

$\frac{d}{dy}2y=2$

Thus, the derivative is 3y^2+2y+2

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Example Question #752 : Other Differential Functions

Find the derivative.

x^6-x^2+x^4

Possible Answers:

6x^5+4x^3-2

x^5+x^3-x

6x^5+4x^3-2x

6x^4+4x^2-2x

Correct answer:

6x^5+4x^3-2x

Explanation:

Use the power rule to find the derivative.

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

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

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

Thus, the derivative is 6x^5+4x^3-2x .

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Example Question #754 : How To Find Differential Functions

Find the derivative.

$\frac{2}{y^4}$

Possible Answers:

$\frac{8}{y^4}$

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

$\frac{-8}{y^4}$

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

Correct answer:

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

Explanation:

Use the quotient rule to find the derivative.

$\frac{(y^4)(0)-(2)(4y^3)}{(y^4)^2}=\frac{-8y^3}{y^8}=\frac{-8}{y^5}$

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Example Question #755 : How To Find Differential Functions

Find the derivative.

3x^2-3x+3

Possible Answers:

6x^2-3x

6x-3

3x-1

Correct answer:

6x-3

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}3x^2=6x$

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

Recall that the derivative of a constant is zero.

Thus, the derivative is 6x-3 .

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Example Question #756 : How To Find Differential Functions

Find the derivative.

$\frac{8}{x}$

Possible Answers:

$\frac{8}{x^2}$

$\frac{8}{x^3}$

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

$\frac{-8}{x^2}$

Correct answer:

$\frac{-8}{x^2}$

Explanation:

Use the quotient rule to find the derivative.

$\frac{x(0)-8(1)}{(x)^2}=\frac{-8}{x^2}$ .

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Example Question #1975 : Calculus

Find the derivative.

4(x^7+2x^3)

Possible Answers:

28x^6+24x^2

7x^6+6x^2

Correct answer:

28x^6+24x^2

Explanation:

Begin by distributing the 4 throughout.

4x^7+8x^3

Now, use the power rule to find the derivative.

$\frac{d}{dx}4x^7=28x^6$

$\frac{d}{dx}8x^3=24x^2$

Thus, the derivative is 28x^6+24x^2

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Example Question #757 : How To Find Differential Functions

Find the derivative.

5x^2+3x+6

Possible Answers:

10x+3

10x^2+3x+6

5x+3

Correct answer:

10x+3

Explanation:

Use the power rule to find this derivative.

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

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

Recall that the derivative of a constant is zero.

Thus, the derivative is 10x+3 .

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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 #931 : Functions

Example Question #751 : How To Find Differential Functions

Example Question #752 : How To Find Differential Functions

Example Question #753 : How To Find Differential Functions

Example Question #752 : Other Differential Functions

Example Question #754 : How To Find Differential Functions

Example Question #755 : How To Find Differential Functions

Example Question #756 : How To Find Differential Functions

Example Question #1975 : Calculus

Example Question #757 : How To Find Differential Functions

All Calculus 1 Resources

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