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Calculus AB : Basic Differentiation Rules

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Example Questions

Example Question #2 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Possible Answers:

$-6e^{x}$

$-6e^{2x}$

$-6e^{-2x}$

$-6e^{-x}$

Correct answer:

$-6e^{-2x}$

Explanation:

We use our rule for finding the derivative of $ae^{nx}$ . We know that $\frac{d}{dx}[ae^{nx}] = (a*n)e^{nx}$ . In this case, a = 3 and n = -2 .

$\frac{d}{dx}[3e^{-2x}] = (3*-2)e^{-2x} = -6e^{-2x}$

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

Which of the following is the correct derivative for f(x) = sin(x) ?

Possible Answers:

cos(x)

-sin(x)

-cos(x)

xsin(x)

Correct answer:

cos(x)

Explanation:

When we are finding the derivative of sin(x) we are finding the rate of change at a certain point/angle of the function sin(x) . If we think about finding the derivative using the graph of sin(x) , we are finding the tangent to sin(x) at a certain point/angle. Thinking about it this way we can see that the derivative of sin(x) is given by the cos of the given angle. The general form for this is $\frac{d}{dx}[asin(nx)] = (a*n)sin(nx)$

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Example Question #4 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function f(x) = 4sin(2x) .

Possible Answers:

f'(x) = -8cos(2x)

f'(x) = 8cos(x)

f'(x) = 8cos(2x)

f'(x) = 8sin(2x)

Correct answer:

f'(x) = 8cos(2x)

Explanation:

We will need to use our rule for finding the derivative of asin(nx) , $\frac{d}{dx}[asin(nx)] = (a*n)sin(nx)$ .

f(x) = 4sin(2x)

f'(x) = (4*2)cos(2x)

f'(x) = 8cos(2x)

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Example Question #5 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Which of the following is the correct derivative for cos(x) ?

Possible Answers:

-sin(x)

-cos(x)

sin(x)

cos(x)

Correct answer:

-sin(x)

Explanation:

To find the derivative of cos(x) , we use the rule $\frac{d}{dx}[acos(nx)] = -(a*n)sin(nx)$ where a,n are constants. We use -sin(x) because -sin(x) is the tangent to the graph of cos(x) at each given point giving us the rate of change at each given angle of the function cos(x) .

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Example Question #6 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Find the derivative of the function f(x) = cos(2x + 3) .

Possible Answers:

f'(x) = 2sin(2x+3)

f'(x) = -2cos(2x+3)

f'(x) = -2sin(2x+3)

Correct answer:

f'(x) = -2sin(2x+3)

Explanation:

We first find the derivative of the quantity of the cosine function. The derivative of 2x+3 is . Now the derivative of cos(x) is -sin(x) . The quantity of the cosine function will stay the same. So now we use the form $\frac{d}{dx}[acos(nx)] = -(a*n)sin(nx)$ to write our derivative.

f(x) = cos(2x+3)

f'(x) = (2)*-sin(2x+3)

f'(x) = -2sin(2x+3)

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Example Question #7 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Which of the following is the correct derivative of ln(x) ?

Possible Answers:

$\frac{1}{x}$

$\frac{1}{ln(x)}$

xln(x)

Correct answer:

$\frac{1}{x}$

Explanation:

The rule for finding the derivative of the natural logarithm is if f(x) = aln(nx) then $\frac{2}{2x} = \frac{1}{x}$ . So if we wanted to find the derivative of $f'(x) = \frac{a*n}{nx}$ we would have ln(2x) .

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Example Question #8 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Possible Answers:

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

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

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

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

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 #9 : 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 #1 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Possible Answers:

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

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

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

f'(x)=6x+2

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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Calculus AB : Basic Differentiation Rules

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #2 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Example Question #4 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #5 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #6 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #7 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #8 : Use Derivatives Of Natural Logs And Advanced Trig Functions

Example Question #9 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

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

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