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Calculus AB : Calculus AB

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45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function $f(x) = -sin(x) + 2e^{3x}$ .

Possible Answers:

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

$f'(x) = cos(x) + 6e^{3x}$

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

$f'(x) = sin(x) + 6e^{3x}$

Correct answer:

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

Explanation:

We know that the derivative of sin(x) is cos(x) and we also know that when we have to find the derivative of e^x we keep the exponent the same but multiply the coefficient of by the coefficient of .

$f(x) = -sin(x) +2e^{3x}$

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

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

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Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Evaluate $lim_{\Delta x \rightarrow 0} \frac{sin(x+\Delta x) -sin(x)}{\Delta x}$

Possible Answers:

$\pi$

cos(x)

tan(x)

Correct answer:

cos(x)

Explanation:

We are able to recognize that this is the definition of derivatives. Once we identify that this is the definition of derivatives we can see that f(x) = sin(x) . So this question is really just asking us to find f'(x) . So our answer here is f'(x) = cos(x) .

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Example Question #3 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function $f(x)=2+\ln(3x^{2}+2x)$ .

Possible Answers:

$f'(x)=\frac{3x^{2}+2x}{6x+2}$

f'(x)=6x+2

$f'(x)=\frac{6x+2}{3x^{2}+2x}$

$f'(x)=\frac{6x+2}{6x}$

Correct answer:

$f'(x)=\frac{6x+2}{3x^{2}+2x}$

Explanation:

We begin with the rule that if $f(x)=a\ln (nx)$ then $f'(x)=\frac{a*n}{nx}$ . a=1 and then we must find the derivative of the quantity within the logarithm function. So if we want the derivative of $3x^{2}+2x$ then we have 6x+2 This will be in the numerator of our derivative.

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

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

$f'(x)=\frac{6x+2}{3x^{2}+2x}$

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Example Question #1 : Apply The Product Rule And Quotient Rule

Which of the following is the Product Rule?

Possible Answers:

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - g'(x)f(x)}{g(x)}$

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

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

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

Correct answer:

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

Explanation:

When two different functions ( f(x) and g(x) ) are being multiplied together and we need to use what we call the product rule to find the derivative. For the product rule you take the derivative of one function at a time while keeping the other constant and sum these together. The formula is: $\frac{d}{dx} [f(x)g(x)]= f'(x)g(x) + g'(x)f(x)$

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Example Question #2 : Apply The Product Rule And Quotient Rule

If we want to find the derivative of the function f(x) = (x^2 - 2)(4x+3) , which of the following would we use?

Possible Answers:

Chain Rule

None of the above

Quotient Rule

Product Rule

Correct answer:

Product Rule

Explanation:

We would use the product rule in this case. Let g(x) = x^2 - 2 and h(x) = 4x + 3 .

We will use the formula $\frac{d}{dx}[g(x)h(x) = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)$

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

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

$\frac{d}{dx}[(x^2 -2)(4x+3)] = 12x^2 + 6x - 8$

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Example Question #3 : Apply The Product Rule And Quotient Rule

Find the derivative (do not solve) of the following function, f(x) = 4xsin(x) .

Possible Answers:

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + sin(x)]$

$\frac{d}{dx}[4xsin(x)] = 4[xsin(x) + sin(x)]$

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + cos(x)]$

$\frac{d}{dx}[4xsin(x)] = 4[xsin(x) + cos(x)]$

Correct answer:

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + sin(x)]$

Explanation:

We will let g(x) = 4x and h(x) = sin(x) .

$\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[4xsin(x)] = 4sin(x) + 4xcos(x)$

$\frac{d}{dx}[4xsin(x)] = 4[xcos(x) + sin(x)]$

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Example Question #4 : Apply The Product Rule And Quotient Rule

Find the derivative (do not solve) of the following function, f(x) =(5x^2 + 8x)(7x - 2) .

Possible Answers:

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x - 36$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x + 36$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 56x^2 + 34x - 16$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 112x^2 + 105x - 36$

Correct answer:

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x - 36$

Explanation:

We will let g(x) = 5x^2 + 8x and h(x) = 7x - 2 .

$\frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x)$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = (10x+8)(7x-2) + (5x^2+8x)(7)$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 70x^2 - 20 + 56x - 16 + 35x^2 + 56x$

$\frac{d}{dx}[(5x^2+8x)(7x-2)] = 105x^2 + 112x - 36$

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Example Question #1 : Apply The Product Rule And Quotient Rule

Which of the following would we use to find the derivative of the function $f(x) = \frac{x^2}{x-2}$

Possible Answers:

Quotient Rule

Product Rule

None of the above

Chain Rule

Correct answer:

Quotient Rule

Explanation:

When trying to find the derivative of the quotient of two functions, we use what is called the quotient rule. The quotient rule is similar to the product rule except you are taking the difference of two products and then dividing them further.

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Example Question #1 : Apply The Product Rule And Quotient Rule

Which of the following is the correct formula for the quotient rule?

Possible Answers:

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

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g'(x) - f(x)g(x)}{g(x)}$

$\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)}$

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

Correct answer:

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

Explanation:

If two functions are differentiable, then so is their quotient. So we use the quotient rule to find the derivative of the entire function.

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Example Question #6 : Apply The Product Rule And Quotient Rule

Find the derivative of the function $f(x) = \frac{e^x}{x^2}$

Possible Answers:

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{x^3 - 2x}{e^2x}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{x^3 - 2x}{e^x}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x^3}$

Correct answer:

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x^3}$

Explanation:

We will need to use the quotient rule here. Let g(x) = e^x and h(x) = x^2 .

$\frac{d}{dx}[\frac{g(x)}{h(x)}] = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{e^x(x^2) - e^x(2x)}{(x^2)^2}$

$\frac{d}{dx} [\frac{e^x}{x^2}] = \frac{x(xe^x - 2e^x)}{x^4}$

$\frac{d}{dx}[\frac{e^x}{x^2}] = \frac{xe^x - 2e^x}{x^3}$

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Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #3 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #1 : Apply The Product Rule And Quotient Rule

Example Question #2 : Apply The Product Rule And Quotient Rule

Example Question #3 : Apply The Product Rule And Quotient Rule

Example Question #4 : Apply The Product Rule And Quotient Rule

Example Question #1 : Apply The Product Rule And Quotient Rule

Example Question #1 : Apply The Product Rule And Quotient Rule

Example Question #6 : Apply The Product Rule And Quotient Rule

All Calculus AB Resources

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