Calculus Examples

$x^{2} + 2 x + 5$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} + 2 x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

Step 1.2

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

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Step 1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 x + 2 \cdot 1 + \frac{d}{d x} [5]$

Step 1.2.3

Multiply $2$ by $1$ .

$2 x + 2 + \frac{d}{d x} [5]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$2 x + 2 + 0$

Step 1.3.2

Add $2 x + 2$ and $0$ .

$2 x + 2$

Step 2

Set the derivative equal to $0$ then solve the equation $2 x + 2 = 0$ .

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Step 2.1

Subtract $2$ from both sides of the equation.

$2 x = - 2$

Step 2.2

Divide each term in $2 x = - 2$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = - 2$ by $2$ .

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 2$ by $2$ .

$x = - 1$

Step 3

Solve the original function $f (x) = x^{2} + 2 x + 5$ at $x = - 1$ .

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Step 3.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = {(- 1)}^{2} + 2 (- 1) + 5$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = 1 + 2 (- 1) + 5$

Step 3.2.1.2

Multiply $2$ by $- 1$ .

$f (- 1) = 1 - 2 + 5$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $2$ from $1$ .

$f (- 1) = - 1 + 5$

Step 3.2.2.2

Add $- 1$ and $5$ .

$f (- 1) = 4$

Step 3.2.3

The final answer is $4$ .

$4$

Step 4

The horizontal tangent line on function $f (x) = x^{2} + 2 x + 5$ is $y = 4$ .

$y = 4$

Step 5