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Calculus 2 : Derivative at a Point

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Example Questions

Example Question #11 : Derivative Defined As Limit Of Difference Quotient

Find the derivative of the following function at x=2 :

$h(x)=2x\sin(x)$

Possible Answers:

$2\sin(2)-4\cos(2)$

$2\sin(2)$

$2\sin(2)+4\cos(2)$

$2\cos(2)$

Correct answer:

$2\sin(2)+4\cos(2)$

Explanation:

The derivative of the function is given by the product rule:

h(x)=f(x)g(x) , h'(x)=f'(x)g(x)+f(x)g'(x)

Simply find the derivative of each function:

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

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

The derivatives were found using the following rules:

$\frac{d}{dx}\sin(x)=\cos(x)$ , $\frac{d}{dx}(x^n)=nx^{n-1}$

Simply evaluate each derivative and the original functions at the point given, using the above product rule.

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Example Question #21 : Derivative At A Point

Find the derivative of the following function about the point (0, 1) :

$f(x)=3x^2+\cos(x)$

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

$f'(x)=6x-\sin(x)$

and was found using the following rules:

$\frac{d}{dx}(x^n)=nx^{n-1}$ , $\frac{d}{dx}\cos(x)= -\sin(x)$

Next, plug in the point we were asked to find the derivative at to finish the problem:

$6(0)-\sin(0)=0$

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Example Question #22 : Derivative At A Point

What is the slope of a function $f(x)=2x^{2}-5x+18$ at the point (3,4) ?

Possible Answers:

Correct answer:

Explanation:

By definition, slope is the first derivative of a given function f(x) .

Since $f(x)=2x^{2}-5x+18$ here, we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)2x^{2-1}-(1)5x^{1-1}+(0)18=4x-5$ .

At (3,4) , x=3 and therefore the slope

f'(3)=4(3)-5=12-5=7 .

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Example Question #106 : Derivative Review

What is the slope of a function $f(x)=4x^{2}+x-11$ at the point (-2,5) ?

Possible Answers:

Correct answer:

Explanation:

By definition, slope is the first derivative of a given function f(x) .

Since $f(x)=4x^{2}+x-11$ here, we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)4x^{2-1}+(1)x^{1-1}-(0)11=8x+1$ .

At the point (-2,5) , we have x=-2 , and so

f'(-2)=8(-2)+1=-16+1=-15 .

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Example Question #102 : Derivative Review

What is the slope of a function $f(x)=7x^{2}-2x+10$ at the point (-1,4) ?

Possible Answers:

None of the above

Correct answer:

Explanation:

By definition, slope is the first derivative of a given function f(x) .

Since $f(x)=7x^{2}-2x+10$ here, we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)7x^{2-1}-(1)2x^{1-1}+(0)10=14x-2$ .

At the point (-1,4) , we have x=-1 , and so

f'(-1)=14(-1)-2=-14-2=-16 .

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Example Question #108 : Derivative Review

Given a function $f(x)=9x^{2}-12x+20$ , what is its slope at the point (4,5) ?

Possible Answers:

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at a given point.

Given

$f(x)=9x^{2}-12x+20$ . we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=(2)9x^{2-1}-(1)12x^{(1-1)}+(0)20=18x-12$ .

Since the -coordinate of (4,5) is , the slope

f'(4)=18(4)-12=72-12=60 .

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Example Question #23 : Derivative At A Point

Given a function $f(x)=-3x^{2}+14x-23$ , what is its slope at the point (-2,-2) ?

Possible Answers:

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at a given point.

Given

$f(x)=-3x^{2}+14x-23$ . we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to derive

$f'(x)=-3(2)x^{2-1}+(1)14x^{1-1}-(0)23=-6x+14$ .

Since the -coordinate of (-2,-2) is , the slope

f'(-2)=-6(-2)+14=12+14=26 .

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Example Question #103 : Derivative Review

Given a function $f(x)=\frac{1}{x^{2}}-3x+2$ , what is its slope at the point (4,1) ?

Possible Answers:

$\frac{97}{32}$

None of the above.

$-\frac{97}{32}$

$\frac{32}{97}$

$-\frac{32}{97}$

Correct answer:

$-\frac{97}{32}$

Explanation:

Slope is defined as the first derivative of a function at a given point. Given $f(x)=\frac{1}{x^{2}}-3x+2$ or $f(x)=x^{-2}-3x+2$ . we can use the Power Rule ( $\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ ) to derive $f'(x)=(-2))x^{-2-1}-(1)3x^{1-1}+(0)2=-2x^{-3}-3=-\frac{2}{x^{3}}-3$ . Since the -coordinate of (4,1) is , the slope $f'(4)=-\frac{2}{4^{3}}-3=-\frac{2}{64}-3=-\frac{1}{32}-3=-\frac{1}{32}-\frac{96}{32}=-\frac{97}{32}$ .

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Example Question #22 : Derivative At A Point

Find the derivative of $y=\left | x-3\right |$ at x=3 .

Possible Answers:

$\pm1$

Does not exist.

Correct answer:

Does not exist.

Explanation:

Split the absolute value into both positive and negative components.

$y=\left | x-3\right | = \pm(x-3)$

Take their derivatives.

$\frac{d}{dx}[y=x-3]=1$

$\frac{d}{dx}[y=-(x-3)]=-1$

At x=3 , there exists a spike in the graph. For spikes, the derivative does not exist under this exception.

The answer is:

$\textup{Does not exist.}$

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Example Question #111 : Derivative Review

What is the slope of a function $f(x)=10x^{2}-25x+12$ at the point (5,7) ?

Possible Answers:

100

Correct answer:

Explanation:

Slope is defined as the first derivative of a function at given point.

We are given the function

$f(x)=10x^{2}-25x+12$ and a point (5,7) , so we need to find the derivative f'(x) and solve for the point's -coordinate.

Using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all nonzero , we can derive

$f'(x)=(2)10x^{2-1}-(1)25x^{1-1}+(0)12=20x-25$ .

Substituting the -coordinate , we have a slope:

f'(5)=20(5)-25=100-25=75 .

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Calculus 2 : Derivative at a Point

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #11 : Derivative Defined As Limit Of Difference Quotient

Example Question #21 : Derivative At A Point

Example Question #22 : Derivative At A Point

Example Question #106 : Derivative Review

Example Question #102 : Derivative Review

Example Question #108 : Derivative Review

Example Question #23 : Derivative At A Point

Example Question #103 : Derivative Review

Example Question #22 : Derivative At A Point

Example Question #111 : Derivative Review

All Calculus 2 Resources

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