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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1361 : Calculus Ii

Find the derivative of the function $y=e^{x^{\frac{1}{2}}+x^2-\ln(x)}$

Possible Answers:

$e^{(\frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x})}$

$(\frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x})e^{x^\frac{1}{2}+x^2-\ln(x)}$

None of the other answers

$\frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x}$

$e^{x^\frac{1}{2}+x^2-\ln(x)}$

Correct answer:

$(\frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x})e^{x^\frac{1}{2}+x^2-\ln(x)}$

Explanation:

The Chain Rule is required here.

$y=e^{x^{\frac{1}{2}}+x^2-\ln(x)}$ . Start

The Chain Rule Proceeds as follows: $f(g(x)) = f'(g(x))\times g'(x)$ . In this case

f(x)= e^x

and

$g(x) = \frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x}$ .

Putting these into the Chain Rule, we get

$y' = e^{x^\frac{1}{2}+x^2-\ln(x)}(\frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x})$

Or the same to say

$y' = (\frac{1}{2}x^{-\frac{1}{2}}+2x-\frac{1}{x})e^{x^\frac{1}{2}+x^2-\ln(x)}$ .

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Example Question #1361 : Calculus Ii

Evaluate the derivative of y = 6x^2 +C , where is any constant.

Possible Answers:

6x^2 +C

None of the other answers

12x

12x+1

Correct answer:

12x

Explanation:

For the 6x^2 term, we simply use the power rule to abtain 12x . Since is a constant (not a variable), we treat it as such. The derivative of any constant (or "stand-alone number") is .

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Example Question #32 : First And Second Derivatives Of Functions

Find the derivative of the function y=7x^2e^x .

Possible Answers:

14x^2(x-1)

$\frac{7}{3}x^3e^x$

14xe^x

7xe^x(x+2)

None of the other answers

Correct answer:

7xe^x(x+2)

Explanation:

We use the Product Rule to find our answer here. The Product Rule formula is (fg)'(x) = f(x)g'(x)+f'(x)g(x) .

Let f(x) = 7x^2 , g(x)= e^x , then we have f'(x) = 14x , g'(x)=e^x .

Putting these into our formula, we have

y' = (7x^2)(e^x)+(e^x)(14x)

=e^x(7x^2+14x)

=7xe^x(x+2) .

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Example Question #1364 : Calculus Ii

What is the derivative of

$f(x)=5\sqrt{x}$ ?

Possible Answers:

$f'(x)=\frac{5}{2\sqrt{x}}$

$f'(x)=\frac{5}{2}\sqrt{x}$

$f'(x)=\frac{5}{\sqrt{x}}$

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

Correct answer:

$f'(x)=\frac{5}{2\sqrt{x}}$

Explanation:

We can find the derivative of

$f(x)=5\sqrt{x}$

using the power rule

$\frac{d}{dx}x^a=ax^{a-1}$

with $a=\frac{1}{2}$

so we have

$f'(x)=5\frac{1}{2}\cdot x^{-\frac{1}{2}}=\frac{5}{2x^{1/2}}=\frac{5}{2\sqrt{x}}$

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Example Question #1365 : Calculus Ii

Find the velocity function given the displacement function:

$f(t)=\pi t^{2/5}$

Possible Answers:

$f'(t)=\frac{\pi}{ t^{3/5}}$

$f'(t)=\frac{2\pi}{ 5t^{2/5}}$

$f'(t)=\frac{2\pi}{ 5t^{3/5}}$

$f'(t)=\frac{5\pi}{ 2t^{3/5}}$

Correct answer:

$f'(t)=\frac{2\pi}{ 5t^{3/5}}$

Explanation:

The derivative of the displacement function is the velocity, so we need to find f'(t) . We can use the power rule

$\frac{d}{dx} x^a=ax^{a-1}$

with $a=\frac{2}{5}$

to get

$f'(t)=\pi\frac{2}{ 5}t^{-3/5}=\frac{2\pi}{ 5t^{3/5}}$

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Example Question #41 : First And Second Derivatives Of Functions

Find the velocity given the displacement function

$f(t)=10 (t-2)^{1/3}$

Possible Answers:

$f'(t)=\frac{10}{ 3(t-2)^{1/3}}$

$f'(t)=\frac{20}{ 3(t-2)^{1/3}}$

$f'(t)=\frac{10}{ 3(t-2)^{2/3}}$

$f'(t)=\frac{20}{ 3(t-2)^{2/3}}$

Correct answer:

$f'(t)=\frac{10}{ 3(t-2)^{2/3}}$

Explanation:

The derivative of the displacement function is the velocity, so we need to find f'(t) . We can use the power rule

$\frac{d}{dx} x^a=ax^{a-1}$

with $a=\frac{1}{3}$

to get

$f'(t)=10\frac{1}{3}(t-2)^{-2/3}=\frac{10}{3(t-2)^{2/3}}$

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Example Question #41 : First And Second Derivatives Of Functions

Find the velocity given the displacement function

$f(t)=5(3t-1)^{3}$

Possible Answers:

f'(t)=45(3t-1)^4

f'(t)=45(3t-1)^2

f'(t)=5(3t-1)^2

f'(t)=15(3t-1)^2

Correct answer:

f'(t)=45(3t-1)^2

Explanation:

The derivative of the displacement function is the velocity, so we need to find f'(t) . We can use the power rule

$\frac{d}{dx} x^a=ax^{a-1}$

with a=3 along with the chain rule to get

$f'(t)=5(3t-1)^{2}\cdot 3\cdot 3=45(3t-1)^2$

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Example Question #43 : First And Second Derivatives Of Functions

Find the velocity given the displacement function

$f(t)=(t^3-1)^{2}$

Possible Answers:

f'(t)=6t^2(t^3-1)^2

f'(t)=3t^2(t^3-1)

f'(t)=6t^3(t^3-1)

f'(t)=6t^2(t^3-1)

Correct answer:

f'(t)=6t^2(t^3-1)

Explanation:

The derivative of the displacement function is the velocity, so we need to find f'(t) . We can use the power rule

$\frac{d}{dx} x^a=ax^{a-1}$

with a=2 along with the chain rule to get

$f'(t)=(t^3-1)^{2}=3t^2\cdot 2(t^3-1)=6t^2(t^3-1)$

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Example Question #42 : First And Second Derivatives Of Functions

Find the velocity given the displacement function

$f(x)=(e^x-1)^{4}$

Possible Answers:

f'(x)=4(e^x-1)^3

f'(x)=4e^x(e^x-1)^3

f'(x)=e^x(e^x-1)^3

$f'(x)=4e^x(e^{2x}-1)^3$

Correct answer:

f'(x)=4e^x(e^x-1)^3

Explanation:

The derivative of the displacement function is the velocity, so we need to find f'(x) . We can use the power rule

$\frac{d}{dx} x^a=ax^{a-1}$

with a=4 along with the chain rule to get

$f'(x)=4\cdot e^x(e^x-1)^3$

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Example Question #45 : First And Second Derivatives Of Functions

Given the displacement function $f(x)=e^{20x+x^5}$ , find the velocity function.

Possible Answers:

$f'(x)=e^{20x+x^5}$

$f'(x)=(20+5x^4)e^{20+5x^4}$

$f'(x)=(20+5x^4)e^{20x+x^5}$

$f'(x)=e^{20+5x^4}$

Correct answer:

$f'(x)=(20+5x^4)e^{20x+x^5}$

Explanation:

To find the velocity of $f(x)=e^{20x+x^5}$ , we need to find the derivative. This can be done with the chain rule:

(g(h(x)))'=g'(h(x))h'(x)

with g(x)=e^x and h(x)=20x+x^5 , so we get

g'(x)=e^x

and

h'(x)=20+5x^4

so we get

$f'(x)=(20+5x^4)e^{20x+x^5}$

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Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1361 : Calculus Ii

Example Question #1361 : Calculus Ii

Example Question #32 : First And Second Derivatives Of Functions

Example Question #1364 : Calculus Ii

Example Question #1365 : Calculus Ii

Example Question #41 : First And Second Derivatives Of Functions

Example Question #41 : First And Second Derivatives Of Functions

Example Question #43 : First And Second Derivatives Of Functions

Example Question #42 : First And Second Derivatives Of Functions

Example Question #45 : First And Second Derivatives Of Functions

All Calculus 2 Resources

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