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Calculus 3 : Derivatives

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Example Questions

Example Question #223 : Calculus 3

Calculate f' (1) of $f(x)= \ln \left(\frac{6x^2}{\left(x+4 \right )} \right )$ .

Possible Answers:

$\frac{9}{5}$

$\ln 6$

$-\frac{2}{15}$

Correct answer:

$\frac{9}{5}$

Explanation:

We begin by rewriting our function in a more convenient way,

$f(x) = \ln \left(\frac{6x^2}{\left(x+4 \right )} \right )= \ln \left(6x^2 \right )- \ln \left(x+4 \right )$ .

This makes taking the derivative a little easier. We can therefore write

$f' (x) = \frac{2}{x}-\frac{1}{x+4}$ , or $f' (1) = 2 - \frac{1}{5} = \frac{9}{5}$ .

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Example Question #41 : Derivatives

Calculate f'(5) of $f(x)= \sin \left(\pi x \right )$ .

Possible Answers:

$\pi$

$\frac{\pi}{2}$

$- \pi$

Correct answer:

$- \pi$

Explanation:

By the chain rule, we can show

$\frac{d}{dx} \sin \left(f(x) \right )= cos \left(f(x) \right ) \cdot f'(x)$ , or in this case,

$\frac{d}{dx} \sin \left( \pi x \right )= \cos \left( \pi x \right ) \cdot \pi\Rightarrow \cos \left(5 \pi \right ) \cdot \pi = -\pi$ .

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Example Question #42 : Derivatives

Find f'(x) given $f(x)=x^{x^2}$

Possible Answers:

$x^{x^2}\left(1+2 \ln x \right )$

$x^{x^2+1}$

$x^{x^2+1}\left(2 \ln x \right )$

$x^{x^2+1}\left(1+2 \ln x \right )$

Correct answer:

$x^{x^2+1}\left(1+2 \ln x \right )$

Explanation:

This derivative must be found using logarithmic differentiation. Consider the following

$\begin{align*} f(x)&=x^{x^2} \\ \log_x f&= x^2\\ \frac{\ln f}{\ln x}&=x^2 \\ \ln f &= x^2 \ln x \\ \frac{f'}{f}&=x+2x \ln x \\ f'&= \left(x + 2x \ln x \right )x^{x^2}= f'(x) = x^{x^2+1}\left(1+2 \ln x \right ) \end{align*}$

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Example Question #43 : Derivatives

Find f'(x) given $f(x)= 7^{3x}$

Possible Answers:

$7^{3x} \cdot \ln 7^3$

$3 \cdot 7^{2x}$

$3 \cdot 7^{3x}$

$\ln 7^3$

Correct answer:

$7^{3x} \cdot \ln 7^3$

Explanation:

This derivative is most easily done by logarithmic differentiation. Consider the following

$\begin{align*} f(x)&=7^{3x} \\ \ln f&=\ln 7 \cdot 3x \\ \frac{f'}{f}&=\ln 7 \cdot 3 \\ f'(x)&=7^{3x} \cdot 3 \ln 7 \\ f'(x)&=7^{3x} \cdot \ln 7^3 \end{align*}$

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Example Question #44 : Derivatives

Find the derivative of: y=-cos(5x+2)

Possible Answers:

-5sin(5x+2)cos(5x+2)

5sin(5x+2)cos(5x+2)

-5sin(5x+2)

5sin(5x+2)

-sin(5x+2)

Correct answer:

5sin(5x+2)

Explanation:

Take the derivative of the cosine function. This will also require chain rule, which is the derivative of the inner function.

$\frac{dy}{dx} = -[-sin(5x+2)] \cdot 5$

The answer is: $\frac{dy}{dx}= 5sin(5x+2)$

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Example Question #45 : Derivatives

Find the derivative: ln(cos(2x))

Possible Answers:

-2tan(2x)

$-\frac{2}{sin(2x)}$

$\frac{1}{sin(2x)}$

$\frac{1}{cos(2x)}$

tan(2x)

Correct answer:

-2tan(2x)

Explanation:

This problem will require multiple chain rule. Take the derivative of natural log, and then apply the inner function of the natural log, cosine, and then also apply the chain rule for .

$\frac{d}{dx}ln(cos(2x)) = \frac{1}{cos(2x)} \cdot -sin(2x) \cdot 2$

Combine this into one term and simplify.

$-\frac{2sin(2x)}{cos(2x)} = -2tan(2x)$

The answer is: -2tan(2x)

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Example Question #46 : Derivatives

Evaluate $\frac{d}{dx}(yx+y)$

Possible Answers:

y+x

y+1

x+1

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is (x+1)y'+y .

Since we are taking the (full) derivative of an expression involving and , we must use implicit differentiation. We proceed as follows

$\frac{d}{dx}(yx+y)$ start.

$=\frac{d}{dx}(yx)+\frac{d}{dx}(y)$

=(xy'+(1)y)+y' . Using $\frac{d}{dx}y=y'$ , and the Product Rule.

=(1+x)y' +y . Factor

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Example Question #47 : Derivatives

Evaluate the limit

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h}$

by interpreting the limit in terms of the definition of the derivative.

Possible Answers:

The limit does not exist.

$\frac{9}{128}$

Correct answer:

$\frac{9}{128}$

Explanation:

The definition of the derivative is

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

Equating

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h}$ , to the definition of the derivative, we have

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h} = \frac{d}{dx}(12x^{12})|_{x=1/2}=144x^{11}|_{x=1/2}$

$=144(\frac{1}{2})^{11}=\frac{144}{2048} =\frac{9}{128}$ .

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Example Question #48 : Derivatives

What is the derivative of $5x^4+2x^2+\frac {3}{x}$ ?

Possible Answers:

$-20x^3+4x-\frac {3}{x^2}$

$20x^3-4x-\frac {3}{x^2}$

$20x^3+4x-\frac {3}{x^2}$

$20x^3+4x-\frac {3}{x}$

Correct answer:

$20x^3+4x-\frac {3}{x^2}$

Explanation:

Step 1: Take the derivative of the first term: 5x^4

$(5x^4)'=20x^{4-1}=20x^3$ via the power rule.
Step 2: Take the derivative of the next term: 2x^2
$(2x^2)'=4x^{2-1}=4x$ , again using the product rule.
Step 3: Take the derivative of the last term $\frac {3}{x}$ .

We will have to use Quotient Rule here.

Find the derivative of f(x) and g(x):

f'(x)=0 , g'(x)=1 .

Use the formula: $\frac {f'(x)g(x)-g'(x)f(x)}{[g(x)]^2}$

$\frac {0(x)-1(3)}{(x)^2}=\frac {-3}{x^2}$ .

Step 4: Put all the final terms from highest to lowest degree...

We get: $20x^3+4x-\frac {3}{x^2}$ .

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Example Question #49 : Derivatives

Find the derivative of

$\sin^n (x^m)$

Possible Answers:

$mn(x^{m-1})\sin^{n-1}(x^m)$

$n(x^{m-1})\sin^{n-1}(x^m)$

$(x^{m-1})\sin^{n-1}(x^m)$

$m^2n^2(x^{m})\sin^{n}(x^m)$

Correct answer:

$mn(x^{m-1})\sin^{n-1}(x^m)$

Explanation:

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Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #223 : Calculus 3

Example Question #41 : Derivatives

Example Question #42 : Derivatives

Example Question #43 : Derivatives

Example Question #44 : Derivatives

Example Question #45 : Derivatives

Example Question #46 : Derivatives

Example Question #47 : Derivatives

Example Question #48 : Derivatives

Example Question #49 : Derivatives

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