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Calculus 3 : Derivatives

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Example Questions

Example Question #241 : Calculus 3

Find the derivative of the following function:

$f(x)=\frac{3x}{(2x-7)^3}$

Possible Answers:

$f'(x)=-\frac{12x+21}{(2x+7)^5}$

$f'(x)=-\frac{12x+21}{(2x+7)^4}$

$f'(x)=\frac{12x-21}{(2x+7)^4}$

$f'(x)=-\frac{33}{(2x+7)^4}$

Correct answer:

$f'(x)=-\frac{12x+21}{(2x+7)^4}$

Explanation:

The easiest way to find the derivative of this function is by using the quotient rule which states that:

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

To find the derivative of the numerator, you must use the power rule which states that:

$\frac{d}{dx}x^n=nx^{n-1}$

To find the derivative of the denominator, you must use the chain rule which states that:

$\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$

After applying all of the stated rules, you get that the derivative of the function is:

$f'(x)=\frac{3(2x-7)^3-18x((2x-7)^2)}{(2x-7)^6}=\frac{(2x-7)^2(3(2x-7)-18x)}{(2x-7)^6}$

After further simplifications, the final answer is:

$f'(x)=-\frac{12x+21}{(2x+7)^4}$

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Example Question #62 : Derivatives

Find the derivative of the following function:

$f(x)=e^{e^{\cot x}}$

Possible Answers:

$f'(x)=-e^{\cot x}(\csc^2 x)$

$f'(x)=e^{\cot x+e^{\cot x}}(\csc^2 x)$

$f'(x)=-e^{e^{\cot x}}(\csc^2 x)$

$f'(x)=-e^{\cot x+e^{\cot x}}(\csc^2 x)$

Correct answer:

$f'(x)=-e^{\cot x+e^{\cot x}}(\csc^2 x)$

Explanation:

The only way to find the derivative of this function is by using the following rule:

$\frac{d}{du}e^u=e^u\cdot u'$

Therefore, part of the derivative will contain the original function, $e^{e^{\cot x}}$ . To solve for the u' portion of the equation, you must take the derivative of the exponent which requires the use of the rule again. The derivative of the exponent is:

$-(e^{\cot x})(\csc^2 x)$

The cosecant comes from the following rule:

$\frac{d}{dx}\cot x=-\csc^2 x$

After combining all of the components in accordance with the rule for differentiating exponential functions, the final answer is:

$f'(x)=-e^{e^{\cot x}}(e^{\cot x})(\csc^2 x)=-e^{\cot x+e^{\cot x}}(\csc^2 x)$

The simplification was made using the following rule:

$x^{a}\cdot x^{b}=x^{a+b}$

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Example Question #61 : Derivatives

Find y'(x) if $y(x)=\frac {x^3+3x+1}{(x+2)^2}$

Possible Answers:

$\frac {x^4+2x^3+12x^2-8x+16}{(x-2)^4}$

$\frac {x^4-2x^3+12x^2-8x+16}{(x-2)^4}$

$\frac {x^4-2x^3+12x^2+8x+16}{(x-2)^4}$

$\frac {4x^4+2x^3+12x^2-8x+16}{(x-2)^4}$

Correct answer:

$\frac {x^4+2x^3+12x^2-8x+16}{(x-2)^4}$

Explanation:

Screen shot 2016 01 07 at 8.17.29 pm

Screen shot 2016 01 07 at 8.17.39 pm

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Example Question #251 : Calculus 3

What is the derivative of y=(2x+3)^4 ?

Possible Answers:

$\frac {(2x+3)^5}{5}$

4x^6(2x+3)

8(2x+3)^3

Correct answer:

8(2x+3)^3

Explanation:

y(x)=(2x+3)^4

Use Chain Rule. Identify f(x) and g(x)

f(x)=2x+3 , g(x)=(2x+3)^4

Find the derivatives:

f'(x)=2, g'(x)=4(2x+3)^3

Use the formula: f'(g(x))(g'(x))

2(4(2x+3)^3)=8(2x+3)^3

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Example Question #61 : Derivatives

What is the derivative of $y=\frac {2x^2+1}{x^3-2x}$

Possible Answers:

None of the Above

$\frac {2x^4+7x^2-2}{(x^3-2x)^2}$

$\frac {-2x^4-7x^2+2}{(x^3-2x)}$

$\frac {-2x^4-7x^2+2}{(x^3-2x)^2}$

$\frac {2x^4+7x^2+2}{(x^3+2x)^2}$

Correct answer:

$\frac {-2x^4-7x^2+2}{(x^3-2x)^2}$

Explanation:

Screen shot 2016 01 08 at 6.11.37 pm

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Example Question #252 : Calculus 3

Find the first derivative of $e^{2x^2}$ .

Possible Answers:

$\large 4xe^{4x}$

$\large -4xe^{2x^2}$

$\large 4xe^{2x^2}$

None of the Above

$\large -4xe^{4x}$

Correct answer:

$\large 4xe^{2x^2}$

Explanation:

Take the derivative of the function...

The derivative of $\large e^x$ is $\large e^x$ , or any power $\large x$ .

We will take the derivative of the exponent and drag it down to the coefficient:

$\large \large (2x^2)'e^{2x^2}=4xe^{2x^2}$

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Example Question #67 : Derivatives

Find the derivative of $\large \large 2+3x-x^3+4x^2+3(7x+1)^4$ .

Possible Answers:

None of the Above

$\large 3+8x-3x^2+84(7x+1)^3$

$\large \large 3-8x-3x^2+84(7x+1)^3$

$\large -3+7x+84(7x+1)^3$

$\large \large 3+7x$

Correct answer:

$\large 3+8x-3x^2+84(7x+1)^3$

Explanation:

Find the derivative of each term:

The derivative of $\large 2$ is $\large 0$ .
The derivative of $\large 3x$ is $\large 3$ .
The derivative of $\large \large -x^3$ is $\large \large -3x^2$ .

The derivative of $\large 4x^2$ is $\large 8x$ .

The derivative of $\large 3(7x+1)^4=3(4)(7)(7x+1)^3=84(7x+1)^3$
Put all of the derivatives together...

$\large 3+8x-3x^2+84(7x+1)^3$

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Example Question #68 : Derivatives

Find the derivative of $\large 2x(4x^2+1)$

Possible Answers:

None of the Above

$\large 0$

$\large 6x^2+3$

$\large 24x^2+2$

$\large 8x^3+2x$

Correct answer:

$\large 24x^2+2$

Explanation:

Separate $\large f(x)$ and $\large g(x)$ :

$\large f(x)=2x$ , $\large g(x)=4x^2+1$
Take the derivative of each function:

$\large f'(x)=2$ , $\large g'(x)=8x$

Use the Product Rule Formula:

$\large [f(x)g(x)]'=f'(x)g(x)+f(x)g'(x)$

$\large 2(4x^2+1)+2x(8x)$

Expand and simplify:

$\large 8x^2+2+16x^2\implies 24x^2+2$

The derivative of the function is $\large 24x^2+2$

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Example Question #253 : Calculus 3

What is the derivative of $\large 2x^2+3x+4$ ?

Possible Answers:

$\large -4x+3$

$\large 4x-3$

None of the Above

$\large -4x-3$

$\large 4x+3$

Correct answer:

$\large 4x+3$

Explanation:

Take the derivative of each term. Use the power rule:

$\large \large (2x^2)'\rightarrow 4x$

$\large (3x)' \rightarrow 3$

$\large (4)' \rightarrow 0$

Add up all the derivatives...

Final Answer is $\large 4x+3$

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Example Question #62 : Derivatives

Find the derivative of $\large y=2x^3-x^4$ .

Possible Answers:

$\large 6x^2+4x^3$

$\large 4x^3-6x^2$

None of the Above

$\large 12x-12x^2$

$\large 6x^2-4x^3$

Correct answer:

$\large 6x^2-4x^3$

Explanation:

Use the power rule to find the derivatives:

$\large (2x^3)'=(3*2)x^{3-1}=6x^2$

$\large (-x^4)'=-4x^3$
Add both derivatives together:

$\large 6x^2-4x^3$

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Calculus 3 : Derivatives

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #241 : Calculus 3

Example Question #62 : Derivatives

Example Question #61 : Derivatives

Example Question #251 : Calculus 3

Example Question #61 : Derivatives

Example Question #252 : Calculus 3

Example Question #67 : Derivatives

Example Question #68 : Derivatives

Example Question #253 : Calculus 3

Example Question #62 : Derivatives

All Calculus 3 Resources

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