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AP Calculus BC : Derivative Defined as Limit of Difference Quotient

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Example Question #14 : Derivative At A Point

Given $f(x)=10x^{2}-9x+1$ , find the value of f'(x) at the point (-2,7) .

Possible Answers:

Correct answer:

Explanation:

Given the function $f(x)=10x^{2}-9x+1$ , we can use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ to find its derivative:

$f'(x)=(2)10x^{2-1}-(1)9x^{1-1}+(0)=20x-9$ .

Plugging in the -value of the point (-2,7) into f'(x) , we get

f'(-2)=20(-2)-9=-40-9=-49 .

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Example Question #15 : Derivative At A Point

Find the derivative of y=(2x+4)(3x+2) at point (1,30) .

Possible Answers:

Correct answer:

Explanation:

Use either the FOIL method to simplify before taking the derivative or use the product rule to find the derivative of the function.

The product rule will be used for simplicity.

y'=(2x+4)(3)+(3x+2)(2) = 6x+12 + 6x+4= 12x+16

Substitute x=1 .

y'(1)= 12(1)+16=28

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Example Question #103 : Derivatives

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

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

Possible Answers:

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

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

$2\cos(2)$

$2\sin(2)$

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 #37 : Derivative At A Point

What is the slope of a function f(x)=-2x^2+5x-3 at the point (0,1) ?

Possible Answers:

Correct answer:

Explanation:

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

Since f(x)=-2x^2+5x-3 , we can use the Power Rule

$\frac{d}{dx}x^n=nx^{n-1}$ for all $n\neq 0$ to determine that

$f'(x)=(2)(-2)x^{2-1}+(1)5x^{1-1}-(0)3=-4x+5$ .

Since we're given a point (0,1) , we can use the x-coordinate to solve for the slope at that point.

Thus,

f'(0)=-4(0)+5=5.

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Example Question #11 : Derivative At A Point

What is the slope of the tangent line to the function

$h(x)=e^{x}ln(x)$

when x = 1?

Possible Answers:

$\frac{1}{e}$

Correct answer:

Explanation:

The slope of the tangent line to a function at a point is the value of the derivative at that point. To calculate the derivative in this problem, the product rule is necessary. Recall that the product rule states that:

$\frac{d}{dx}(m(x)n(x))=m'(x)n(x)+m(x)n'(x)$ .

In this example, $m(x)=e^x \ and\ n(x)=ln(x).$

Therefore,

$m'(x)=e^x \ and\ n'(x)=\frac{1}{x}$ , and

$\frac{d}{dx}(e^xln(x))=e^xln(x)+e^x\cdot\frac{1}{x}$

At x = 1, this dervative has the value

$e^1ln(1)+e^1\cdot\frac{1}{1}=0+e=e$ .

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Example Question #52 : Derivative At A Point

Find f'(0) for

$f(x)=x^3\ln(x+1)$

Possible Answers:

$\infty$

Correct answer:

Explanation:

In order to find the derivative, we need to find f'(x) . We can find this by remembering the product rule and knowing the derivative of natural log.

Product Rule:

[f(x)g(x)]'=f'(x)g(x)+f(x)g'(x) .

Derivative of natural log:

$f(x)=\ln(g(x))$

$f'(x)=\frac{g'(x)}{g(x)}$

Now lets apply this to our problem.

$f(x)=x^3\ln(x+1)$

$f'(x)=3x^2\ln(x+1)+\frac{x^3}{x+1}$

$f'(0)=3(0)^2\ln(0+1)+\frac{0^3}{0+1}=0$

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Example Question #57 : Derivative At A Point

Find the second derivative of the following function at x=0 :

$e^{2x}+2x\cos(3x)$

Possible Answers:

Correct answer:

Explanation:

To find the second derivative of the function, we first must find the first derivative of the function:

$f'(x)=2e^{2x}+2\cos(3x)-6x\sin(3x)$

The derivative was found using the following rules:

$\frac{d}{dx}\cos(x)=-\sin(x)$ , $\frac{d}{dx}e^u=e^u\frac{du}{dx}$ , $\frac{d}{dx}(x^n)=nx^{n-1}$ , $\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{d}{dx}f(x)g(x)=f'(x)g(x)+f(x)g'(x)$

The second derivative is simply the derivative of the first derivative function, and is equal to:

$f''(x)=4e^{2x}-12\sin(3x)-18x\cos(3x)$

One more rule used in combination with some of the ones above is:

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

To finish the problem, plug in x=0 into the above function to get an answer of .

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Example Question #61 : Derivatives

Calculate the derivative of $f(x)=\frac{x^3}{3}-2x^2+x-3$ at the point x=3 .

Possible Answers:

Correct answer:

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of f(x)

Then, replace the value of x with the given point and evaluate

For example, if x=a , then we are looking for the value of f'(x=a) , or the derivative of f(x) at x=a .

$f(x)=\frac{x^3}{3}-2x^2+x-3$

Calculate f'(x)

Derivative rules that will be needed here:

Derivative of a constant is 0. For example, $\frac{\mathrm{d} }{\mathrm{d} t} 5 = 0$
Taking a derivative on a term, or using the power rule, can be done by doing the following: $\frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}$

f'(x)=x^2 - 4x + 1

Then, plug in the value of x and evaluate

f'(x=3)=3^2-4*3+1=-2

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

If $f(x)=\ln(2x)$ , which of the following limits equals f'(e+1) ?

Possible Answers:

$\lim_{h \to 0} \frac{\ln(e+1+h)-\ln(e+1)}{e+1}$

$\lim_{h \to 0} \frac{\ln(e+1+h)-\ln(e+1)}{h}$

$\lim_{h \to 0} \frac{\ln(1+h)-\ln(h)}{h}$

$\lim_{h \to \infty} \frac{\ln(e+1+h)-\ln(e+1)}{e+1}$

$\lim_{h \to \infty} \frac{\ln(e+1+h)-\ln(e+1)}{h}$

Correct answer:

$\lim_{h \to 0} \frac{\ln(e+1+h)-\ln(e+1)}{h}$

Explanation:

The equation for the derivative at a point is given by

$f'(a)=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$ .

By substituting a = e+1 , $f = \ln(2x)$ , we obtain

$f'(e+1)=\lim_{h \to 0} \frac{\ln(e+1+h)-\ln(e+1)}{h}.$

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AP Calculus BC : Derivative Defined as Limit of Difference Quotient

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #14 : Derivative At A Point

Example Question #15 : Derivative At A Point

Example Question #103 : Derivatives

Example Question #37 : Derivative At A Point

Example Question #11 : Derivative At A Point

Example Question #52 : Derivative At A Point

Example Question #57 : Derivative At A Point

Example Question #61 : Derivatives

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

All AP Calculus BC Resources

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