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Calculus AB : Use Derivatives of Natural Logs and Advanced Trig Functions

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Example Questions

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

Which of the following is the derivative of e^x ?

Possible Answers:

e^x

$e^{-x}$

xe^x

Correct answer:

e^x

Explanation:

When finding the derivative of $ae^{nx}$ we use the power rule by multiplying the coefficient of by the coefficient of . The form of this looks like $\frac{d}{dx}[ae^{nx}] = a*ne^{nx}$ . This is actually a case of using the Chain Rule, but that will be covered in later topics. So in this case the coefficient of is and the coefficient of is also . So the $\frac{d}{dx}[e^x] = e^x$ .

If we were to find the derivative of $e^{2x}$ , then $\frac{d}{dx}[e^{2x}] = 2e^{2x}$ .

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Example Question #1 : 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 #2 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Possible Answers:

xsin(x)

-sin(x)

-cos(x)

cos(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 #3 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Possible Answers:

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

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

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

f'(x) = 8cos(x)

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

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

Possible Answers:

cos(x)

-cos(x)

-sin(x)

sin(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 #5 : Use Derivatives Of Natural Logs And Advanced Trig Functions

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

Possible Answers:

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

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

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

f'(x) = -2cos(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 #21 : Basic Differentiation Rules

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

Possible Answers:

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

xln(x)

$\frac{1}{x}$

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 #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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Calculus AB : Use Derivatives of Natural Logs and Advanced Trig Functions

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

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

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

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

Example Question #21 : Basic Differentiation Rules

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

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