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Calculus 1 : Functions

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10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #723 : How To Find Differential Functions

Find the derivative.

x^4-x^3+x

Possible Answers:

x^3-x^2

4x^3-3x^2+1

x^3+x^2+1

x^4-x^3+x

Correct answer:

4x^3-3x^2+1

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}x^4=4x^3$

$\frac{d}{dx}-x^3=-3x^2$

$\frac{d}{dx}x=1$

Thus, the derivative is 4x^3-3x^2+1 .

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Example Question #723 : How To Find Differential Functions

Find the derivative:

f(x)= ln(2x+e^x)

Possible Answers:

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

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

$f'(x)= \frac{ln(2+e^x)}{2x+e^x}$

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

Answer not listed

Correct answer:

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

Explanation:

If f(x) = C(constant) , then the derivative is f'(x)= 0 .

If f(x)= x^C , then the derivative is $f'(x)= Cx^{C-1}$ .

If f(x)= ln(x) , then the derivative is $f'(x)= \frac{1}{x}$ .

If f(x)= e^x , the the derivative is f'(x)= e^x .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If F(x)= (f*g)(x) , then the derivative is F'(x)= f'(g(x)) g'(x) . This is known as the chain rule.

In this case, we must find the derivative of the following: f(x)= ln(2x+e^x)

That is done by doing the following: $f'(x)= \frac{1}{2x+e^x} (2x^{1-1}+e^x)$

Therefore, the answer is: $f'(x)= \frac{2+e^x}{2x+e^x}$

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Example Question #724 : How To Find Differential Functions

Find the derivative of the following function:

$f(x)= 4e^{2x} + 3x^4$

Possible Answers:

f'(x)= 8 + 12x

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

$f'(x)= 8e^{2x} + 12x^{3}$

Answer not listed

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

Correct answer:

$f'(x)= 8e^{2x} + 12x^{3}$

Explanation:

If f(x) = C(constant) , then the derivative is f'(x)= 0 .

If f(x)= x^C , then the derivative is $f'(x)= Cx^{C-1}$ .

If f(x)= ln(x) , then the derivative is $f'(x)= \frac{1}{x}$ .

If f(x)= e^x , the the derivative is f'(x)= e^x .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If F(x)= (f*g)(x) , then the derivative is F'(x)= f'(g(x)) g'(x) . This is known as the chain rule.

In this case, we must find the derivative of the following: $f(x)= 4e^{2x} + 3x^4$

That is done by doing the following: $f'(x)= 4e^{2x}(2) + (4)3x^{4-1}$

Therefore, the answer is: $f'(x)= 8e^{2x} + 12x^{3}$

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Example Question #911 : Functions

Find the first derivative of x^3-15x^2+75x-45 .

Possible Answers:

3x^2-30x+30

3x^2-30x+75

x^2-15x+75

3x^2+6x-12

x^2-15x+30

Correct answer:

3x^2-30x+75

Explanation:

We need to differentiate term by term, applying the power rule,

$(x^n)'=nx^{n-1}$

This gives us

$\begin{align*} (x^3-15x^2+75x-45)' &= 3x^2-30x+75 \end{align*}$

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Example Question #912 : Functions

Find the first derivative of x^3-3x^2-2 .

Possible Answers:

3x^2-6x-2

3x^2-6x

x^2-3x-2

x^2-3x

None of the other answers.

Correct answer:

3x^2-6x

Explanation:

We need to differentiate term by term, applying the power rule,

$(x^n)'=nx^{n-1}$

This gives us

$\begin{align*} (x^3-3x^2-2)' &= 3x^2-6x \end{align*}$

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Example Question #727 : How To Find Differential Functions

Find the dervivative: f(x)= (x^4 + ln(x) - 3)^2

Possible Answers:

$f'(x)= (x^4 + ln(x) - 3) (4x^{3} + \frac{1}{x} )$

Answer not listed

$f'(x)= 2(x^4 + ln(x) - 3)^2 (4x^{3} + \frac{1}{x} )$

$f'(x)= 2(x^4 + ln(x) - 3) (4x^{3} + \frac{1}{x} )$

$f'(x)= 2(x^4 + ln(x) - 3) (4x^{3} + \frac{1}{x} )^2$

Correct answer:

$f'(x)= 2(x^4 + ln(x) - 3) (4x^{3} + \frac{1}{x} )$

Explanation:

If f(x) = C(constant) , then the derivative is f'(x)= 0 .

If f(x)= x^C , then the derivative is $f'(x)= Cx^{C-1}$ .

If f(x)= ln(x) , then the derivative is $f'(x)= \frac{1}{x}$ .

If f(x)= e^x , the the derivative is f'(x)= e^x .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If F(x)= (f*g)(x) , then the derivative is F'(x)= f'(g(x)) g'(x) . This is known as the chain rule.

In this case, we must find the derivative of the following: f(x)= (x^4 + ln(x) - 3)^2

That is done by doing the following: $f'(x)= (2)(x^4 + ln(x) - 3)^{(2-1)} (4x^{(4-1)} + \frac{1}{x} - 0)$

Therefore, the answer is: $f'(x)= 2(x^4 + ln(x) - 3) (4x^{3} + \frac{1}{x} )$

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Example Question #728 : How To Find Differential Functions

Find the derivative: $f(x)= \sin(2x^4 - e^{2x})$

Possible Answers:

$f'(x)= -\cos(2x^4 - e^{2x}) (8x^{3} - e^{2x})$

$f'(x)= \sin(2x^4 - e^{x}) (8x^{3} - e^{x})$

$f'(x)= \cos(2x^4 - e^{2x}) (8x^{3} - 2e^{2x})$

Answer not listed

$f'(x)= -\cos(8x^3 - e^{2x}) (8x^{3} - e^{2x})$

Correct answer:

$f'(x)= \cos(2x^4 - e^{2x}) (8x^{3} - 2e^{2x})$

Explanation:

If f(x) = C(constant) , then the derivative is f'(x)= 0 .

If f(x)= x^C , then the derivative is $f'(x)= Cx^{C-1}$ .

If f(x)= ln(x) , then the derivative is $f'(x)= \frac{1}{x}$ .

If f(x)= e^x , the the derivative is f'(x)= e^x .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If F(x)= (f*g)(x) , then the derivative is F'(x)= f'(g(x)) g'(x) . This is known as the chain rule.

In this case, we must find the derivative of the following: $f(x)= \sin(2x^4 - e^{2x})$

That is done by doing the following: $f'(x)= \cos(2x^4 - e^{2x}) ((4)2x^{(4-1)} - e^{2x} (2))$

Therefore, the answer is: $f'(x)= \cos(2x^4 - e^{2x}) (8x^{3} - 2e^{2x})$

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Example Question #731 : Other Differential Functions

Find the derivative: $f(x)= e^{4x^5 + ln(x)}$

Possible Answers:

Answer not listed

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

$f'(x)= e^{20x^{4} + \frac{1}{x}} (20x^{4} + \frac{1}{x})$

$f'(x)= e^{4x^5 + ln(x)} (20x^{4} + \frac{1}{x})$

$f'(x)= e^{4x^5 + ln(x)} (4x^{5} + \frac{1}{x})$

Correct answer:

$f'(x)= e^{4x^5 + ln(x)} (20x^{4} + \frac{1}{x})$

Explanation:

If f(x) = C(constant) , then the derivative is f'(x)= 0 .

If f(x)= x^C , then the derivative is $f'(x)= Cx^{C-1}$ .

If f(x)= ln(x) , then the derivative is $f'(x)= \frac{1}{x}$ .

If f(x)= e^x , the the derivative is f'(x)= e^x .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If F(x)= (f*g)(x) , then the derivative is F'(x)= f'(g(x)) g'(x) . This is known as the chain rule.

In this case, we must find the derivative of the following: $f(x)= e^{4x^5 + ln(x)}$

That is done by doing the following: $f'(x)= e^{4x^5 + ln(x)} ((5)4x^{(5-1)} + \frac{1}{x})$

Therefore, the answer is: $f'(x)= e^{4x^5 + ln(x)} (20x^{4} + \frac{1}{x})$

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Example Question #732 : Other Differential Functions

Find the derivative: $f(x)= -\sin( 2x+ e^{2x})$

Possible Answers:

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

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

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

Answer not listed

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

Correct answer:

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

Explanation:

If f(x) = C(constant) , then the derivative is f'(x)= 0 .

If f(x)= x^C , then the derivative is $f'(x)= Cx^{C-1}$ .

If f(x)= ln(x) , then the derivative is $f'(x)= \frac{1}{x}$ .

If f(x)= e^x , the the derivative is f'(x)= e^x .

If $f(x)= \sin x$ , then the derivative is $f'(x)= \cos x$ .

There are many other rules for the derivatives for trig functions.

If F(x)= (f*g)(x) , then the derivative is F'(x)= f'(g(x)) g'(x) . This is known as the chain rule.

In this case, we must find the derivative of the following: $f(x)= -\sin( 2x+ e^{2x})$

That is done by doing the following: $f'(x)= -\cos( 2x+ e^{2x}) ((1)2x^{1-1} + e^{2x} (2))$

Therefore, the answer is: $f'(x)= -\cos( 2x+ e^{2x}) (2 + 2e^{2x})$

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Example Question #913 : Functions

Find the derivative.

x^9+2x^8-4

Possible Answers:

x^8+x^7

9x^8+16x^7-1

9x^8+16x^7

9x^8+16x^7-4

Correct answer:

9x^8+16x^7

Explanation:

Use the power rule to find the derivative.

$\frac{d}{dx}x^9=9x^8$

$\frac{d}{dx}2x^8=16x^7$

$\frac{d}{dx}-4=0$

Thus, the derivative is 9x^8+16x^7 .

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Calculus 1 : Functions

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #723 : How To Find Differential Functions

Example Question #723 : How To Find Differential Functions

Example Question #724 : How To Find Differential Functions

Example Question #911 : Functions

Example Question #912 : Functions

Example Question #727 : How To Find Differential Functions

Example Question #728 : How To Find Differential Functions

Example Question #731 : Other Differential Functions

Example Question #732 : Other Differential Functions

Example Question #913 : Functions

All Calculus 1 Resources

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