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Precalculus : Introductory Calculus

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12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #28 : Find The First Derivative Of A Function

Use any method to find the first derivative of the function where $y=(4\ln(2x))^3$ .

Possible Answers:

$y'=3(4x^5\ln(2x))^2$

$y'=1152x^{14}(\ln(2x))^2$

$y'=45x^{13}(\ln(2x))^2$

$y'=\frac{12(4ln(2x))^2}{x}$

Correct answer:

$y'=\frac{12(4ln(2x))^2}{x}$

Explanation:

You can use the Chain Rule here,

$[f(g(x))]'=f'(g(x))\cdot g'(x)$

$y=(4\ln(2x))^3$

In our case we get,

$y'=3(4ln(2x))^2\cdot \left(4\frac{1}{2x}\cdot 2\right)$ .

Remember the derivative of natural log is

$ln(u)\rightarrow \frac{1}{u}\cdot \frac{du}{dx}$ .

Simplifying further we get our final answer.

$y'=\frac{12(4ln(2x))^2}{x}$

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Example Question #29 : Find The First Derivative Of A Function

What is the first derivative of f(x) = x^2 + 5x + 2 ?

Possible Answers:

x + 5

2x + 5

2x + 7

None of the answers listed

Correct answer:

2x + 5

Explanation:

To find the derivative of a polynomial function, remember to use the power rule. Take the derivative of each term individually, making the power the new coefficient and then decreasing the power by one. Let us show how this works for each term.

$x^2\rightarrow 2x^{(2-1)} \rightarrow 2x$

$5x \rightarrow5x^{1-1} = 5x^0 = 5$

Since is a constant, we know that the derivative of a constant is just .

Thus, our entire expression is 2x + 5 .

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Example Question #31 : Find The First Derivative Of A Function

Find the derivative: f(x)=6x^4+2x^3-x^2+5

Possible Answers:

f'(x)=24x^3+6x^2-2x+5x

Answer is not present.

f'(x)=6x^3+3x^2-2x+5

f'(x)=24x^3+6x^2-2x

f'(x)=6x^3+3x^2-2x

Correct answer:

f'(x)=24x^3+6x^2-2x

Explanation:

f(x)=6x^4+2x^3-x^2+5

The derivative of this function, known as f'(x) (read as f prime of x), can be found by using the Power rule of derivatives on each term in the function. Power rule: $x^n=nx^{n-1}$ . If there is a value in front of x we multiply it by the "n" we carried over. For example taking the derivative of 2x^4=8x^3 .

Using the Power rule:

f'(x)=24x^3+6x^2-2x

Note that the derivative of any constant is simply 0.

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Example Question #32 : Find The First Derivative Of A Function

Find the first derivative of the function

$f(x)=x^{2}cos(2x)$

Possible Answers:

$f'(x)=2xcos(2x)-2x^{2}sin(2x)$

$f'(x)=-2xsin(2x)-x^{2}cos(2x)$

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

$f'(x)=x^{2}cos(2x)-2sin(2x)$

f'(x)=-4xsin(2x)

Correct answer:

$f'(x)=2xcos(2x)-2x^{2}sin(2x)$

Explanation:

Use the product rule to find the first derivative.

h(x)=f(x)g(x)

h'(x)=f'(x)g(x)+g'(x)f(x)

Here, let $f(x)=x^{2}$ and g(x)=cos(2x)

$h'(x)=(x^{2})'cos(2x)+x^{2}(cos(2x))'$

$h'(x)=2xcos(2x)+(-2x^{2}sin(2x))$

$h'(x)=2xcos(2x)-2x^{2}sin(2x)$

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Example Question #32 : Find The First Derivative Of A Function

Find the first derivative of the function

$f(x)=\sqrt{x^{2}+4x}$ .

Possible Answers:

$f'(x)=(2x+4)\sqrt{x^{2}+4x}$

$f'(x)=\frac{2x+4}\sqrt{x^{2}+4x}{}$

$f'(x)=\frac{x+2}{\sqrt{x^{2}+4x}}$

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

$f'(x)=2x\sqrt{x+4}$

Correct answer:

$f'(x)=\frac{x+2}{\sqrt{x^{2}+4x}}$

Explanation:

Using the power rule, $f(x)=\sqrt{x}=x^{\frac{1}{2}}$

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

Apply the Chain Rule.

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

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

$f'(x)=\frac{x+2}{\sqrt{x^{2}+4x}}$

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Example Question #34 : Find The First Derivative Of A Function

f(x)=(x^4-2x-1) and g(x)=3x-5 . Find the derivative of h(x)=f(x)g(x) .

Possible Answers:

f'(x)=15x^4-20x^2-12x+7

f'(x)=15x^4-20x^3+7

f'(x)=15x^3-20x^3+12x+7

f'(x)=15x^4-20x^3-12x+7

f'(x)=15x^4-20x^3-12x-7

Correct answer:

f'(x)=15x^4-20x^3-12x+7

Explanation:

Since we have the product of two different expressions we will use the product rule: f'(x)g(x)+f(x)g'(x) .

(x^4-2x-1)(3x-5)

Using the product rule:

(4x^3-2)(3x-5)+(x^4-2x-1)*3

Combine and simplify:

12x^4-20x^3-6x+10+3x^4-6x-3

h'(x)=15x^4-20x^3-12x+7

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Example Question #35 : Find The First Derivative Of A Function

What is the derivative of the following function:

y=tan(x)-2cos(x)

with respect to .

Possible Answers:

sec^2(t)+2sin(t)

$(sec^2(x)+2sin(x))\cdot t$

(tan(x)-2cos(x))t

Correct answer:

Explanation:

This question is meant to test your understanding of derivatives.

Since the term tan(x)-2cos(x) is completely unrelated to the variable ,

$\frac{\mathrm{d} }{\mathrm{d} t}(tan(x)-2cos(x))=0$ .

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Example Question #121 : Introductory Calculus

Determine the first derivative of y=2x^5 .

Possible Answers:

10x^5

7x^4

10x

10x^4

7x^5

Correct answer:

10x^4

Explanation:

Use the power rule to solve this derivative.

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

$y'=5\cdot 2x^{5-1} = 10x^4$

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Example Question #37 : Find The First Derivative Of A Function

Calculate the derivative of $y= 5\pi + 2$ .

Possible Answers:

$\frac{2}{5}$

$\frac{5}{2}$

Correct answer:

Explanation:

The derivative of constants, or numbers, are zero. The term $5\pi + 2$ has no variables and will simplify into some number if the $\pi$ was replaced with 3.14 .

The result of $5\pi + 2$ will be a constant. Therefore, the derivative, or slope, of $y= 5\pi + 2$ is zero.

The answer is .

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Example Question #38 : Find The First Derivative Of A Function

Find the derivative of: y=(2x+2)^2

Possible Answers:

4x+4

8x+8

4x+8

8x+4

8x+12

Correct answer:

8x+8

Explanation:

To find the derivative of this function, use the power rule and the chain rule.

The power rule is: $\frac{d}{dx} x^n = nx^{n-1}$

The chain rule is to take the derivative of the inner function.

Apply the power rule for the function and the chain rule. The derivative of 2x+2 is .

$y'=2(2x+2) \cdot 2 = 4(2x+2) = 8x+8$

The answer is: 8x+8

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Precalculus : Introductory Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #28 : Find The First Derivative Of A Function

Example Question #29 : Find The First Derivative Of A Function

Example Question #31 : Find The First Derivative Of A Function

Example Question #32 : Find The First Derivative Of A Function

Example Question #32 : Find The First Derivative Of A Function

Example Question #34 : Find The First Derivative Of A Function

Example Question #35 : Find The First Derivative Of A Function

Example Question #121 : Introductory Calculus

Example Question #37 : Find The First Derivative Of A Function

Example Question #38 : Find The First Derivative Of A Function

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