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Precalculus : Find the First Derivative of a Function

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Example Questions

Example Question #132 : Introductory Calculus

Find the first derivative of the following function:

f(x)=5x-7x^2-2

Possible Answers:

f'(x)=5+14x

f'(x)=-14x+5

f'(x)=5x-7x^2-2

f'(x)=5x+14

Correct answer:

f'(x)=-14x+5

Explanation:

To solve, simply differentiate using the power rule for differentiation as outline below.

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

Thus,

$f'(x)=5x^{1-1}-7x^{2-1}$

Notice that our constant term disappeared because the derivative of a constant is zero.

Simplifying the above equation, we get:

f'(x)=5x^0-7x^1=5-7x

Thus, our answer is:

f'(x)=5-7x

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Example Question #751 : Pre Calculus

Find the first derivative of the following equation:

f(x)=x^2+77x-1

Possible Answers:

2x+77

2x+7

Correct answer:

2x+77

Explanation:

To solve, simply use the power rule as outline below:

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

Thus,

$f'(x)=2x^{2-1}+77x^{1-1}=2x+77$

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Example Question #41 : Find The First Derivative Of A Function

Find the first derivative of the following function:

f(x)=(2x+1)(x^2-3)

Possible Answers:

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

f(x)=(2x-1)(2x)-(2)(x^2-3)

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

Correct answer:

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

Explanation:

To solve, you must use the product rule as outline below.

Product rule: (g(x)*h(x))'=g(x)h'(x)+g'(x)h(x)

$g(x)=(2x+1)\rightarrow g'(x)=2$

$h(x)=(x^2-3)\rightarrow h'(x)=2x$

Thus,

f'(x)=(2x+1)(2x)+(2)(x^2-3)

Distribute the 2x and 2.

f'(x)=4x^2+2x+2x^2-6

Combine like terms to simplify.

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

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Example Question #758 : Pre Calculus

Find the first derivative of the function $f(x) = \frac{1}{x}$ .

Possible Answers:

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

f(x) = ln(x)

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

Correct answer:

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

Explanation:

An equivalent form of writing $f(x) = \frac{1}{x}$ is $f(x) = x^{-1}$ . The derivative of an exponential power x^b is $bx^{b-1}$ , so the derivative of $f(x) = x^{-1}$ is $f'(x) = -x^{-2}$ , or $f'(x) = \frac{-1}{x^2}$ .

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Example Question #759 : Pre Calculus

Find the first derivative: 12x^4-3x^3+2x^2-12-6x-9x^3

Possible Answers:

None of the other answers.

12x^3-3x^2+4x-18

48x^3-36x^2+4x+18

48x^3-36x^2+4x-6

48x^3-36x^2+4x-18

Correct answer:

48x^3-36x^2+4x-6

Explanation:

12x^4-3x^3+2x^2-12-6x-9x^3

Simplify first:

12x^4-12x^3+2x^2-12-6x

Use the power rule on each term:

Power rule: $x^a=ax^{a-1}$

48x^3-36x^2+4x-6

Note that constants become zero.

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Precalculus : Find the First Derivative of a Function

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #132 : Introductory Calculus

Example Question #751 : Pre Calculus

Example Question #41 : Find The First Derivative Of A Function

Example Question #758 : Pre Calculus

Example Question #759 : Pre Calculus

All Precalculus Resources

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