Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of

$x^{2} + 2 x + 1$ with respect to

$x$ is

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

Step 1.1.1.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [1]$

Step 1.1.2

Evaluate

$\frac{d}{d x} [2 x]$ .

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Step 1.1.2.1

Since

$2$ is constant with respect to

$x$ , the derivative of

$2 x$ with respect to

$x$ is

$2 \frac{d}{d x} [x]$ .

$2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$2 x + 2 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.2.3

Multiply

$2$ by

$1$ .

$2 x + 2 + \frac{d}{d x} [1]$

Step 1.1.3

Differentiate using the Constant Rule.

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Step 1.1.3.1

Since

$1$ is constant with respect to

$x$ , the derivative of

$1$ with respect to

$x$ is

$0$ .

$2 x + 2 + 0$

Step 1.1.3.2

Add

$2 x + 2$ and

$0$ .

$f' (x) = 2 x + 2$

Step 1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$2 x + 2$ .

$2 x + 2$

Step 2

Set the first derivative equal to

$0$ then solve the equation

$2 x + 2 = 0$ .

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Step 2.1

Set the first derivative equal to

$0$ .

$2 x + 2 = 0$

Step 2.2

Subtract

$2$ from both sides of the equation.

$2 x = - 2$

Step 2.3

Divide each term in

$2 x = - 2$ by

$2$ and simplify.

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Step 2.3.1

Divide each term in

$2 x = - 2$ by

$2$ .

$\frac{2 x}{2} = \frac{- 2}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of

$2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 2}{2}$

Step 2.3.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 2}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide

$- 2$ by

$2$ .

$x = - 1$

Step 3

The values which make the derivative equal to

$0$ are

$- 1$ .

$- 1$

Step 4

After finding the point that makes the derivative

$f' (x) = 2 x + 2$ equal to

$0$ or undefined, the interval to check where

$f (x) = x^{2} + 2 x + 1$ is increasing and where it is decreasing is

$(- \infty, - 1) \cup (- 1, \infty)$ .

$(- \infty, - 1) \cup (- 1, \infty)$

Step 5

Substitute a value from the interval

$(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable

$x$ with

$- 2$ in the expression.

$f' (- 2) = 2 (- 2) + 2$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply

$2$ by

$- 2$ .

$f' (- 2) = - 4 + 2$

Step 5.2.2

Add

$- 4$ and

$2$ .

$f' (- 2) = - 2$

Step 5.2.3

The final answer is

$- 2$ .

$- 2$

Step 5.3

$x = - 2$ the derivative is

$- 2$ . Since this is negative, the function is decreasing on

$(- \infty, - 1)$ .

Decreasing on

$(- \infty, - 1)$ since

$f' (x) < 0$

Decreasing on

$(- \infty, - 1)$ since

$f' (x) < 0$

Step 6

Substitute a value from the interval

$(- 1, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable

$x$ with

$0$ in the expression.

$f' (0) = 2 (0) + 2$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply

$2$ by

$0$ .

$f' (0) = 0 + 2$

Step 6.2.2

Add

$0$ and

$2$ .

$f' (0) = 2$

Step 6.2.3

The final answer is

$2$ .

$2$

Step 6.3

$x = 0$ the derivative is

$2$ . Since this is positive, the function is increasing on

$(- 1, \infty)$ .

Increasing on

$(- 1, \infty)$ since

$f' (x) > 0$

Increasing on

$(- 1, \infty)$ since

$f' (x) > 0$

Step 7

List the intervals on which the function is increasing and decreasing.

Increasing on:

$(- 1, \infty)$

Decreasing on:

$(- \infty, - 1)$

Step 8

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