Calculus Examples

$f (x) = x^{2} + 11 x - 15$

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 $x^{2} + 11 x - 15$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 15]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 15]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [11 x] + \frac{d}{d x} [- 15]$

Step 1.2

Evaluate $\frac{d}{d x} [11 x]$ .

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Step 1.2.1

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$2 x + 11 \frac{d}{d x} [x] + \frac{d}{d x} [- 15]$

Step 1.2.2

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

$2 x + 11 \cdot 1 + \frac{d}{d x} [- 15]$

Step 1.2.3

Multiply $11$ by $1$ .

$2 x + 11 + \frac{d}{d x} [- 15]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15$ with respect to $x$ is $0$ .

$2 x + 11 + 0$

Step 1.3.2

Add $2 x + 11$ and $0$ .

$2 x + 11$

Step 2

Set the derivative equal to $0$ then solve the equation $2 x + 11 = 0$ .

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Step 2.1

Subtract $11$ from both sides of the equation.

$2 x = - 11$

Step 2.2

Divide each term in $2 x = - 11$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = - 11$ by $2$ .

$\frac{2 x}{2} = \frac{- 11}{2}$

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.

$\frac{2 x}{2} = \frac{- 11}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{2}$

Step 3

Solve the original function $f (x) = x^{2} + 11 x - 15$ at $x = - \frac{11}{2}$ .

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Step 3.1

Replace the variable $x$ with $- \frac{11}{2}$ in the expression.

$f (- \frac{11}{2}) = {(- \frac{11}{2})}^{2} + 11 (- \frac{11}{2}) - 15$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1

Apply the product rule to $- \frac{11}{2}$ .

$f (- \frac{11}{2}) = {(- 1)}^{2} {(\frac{11}{2})}^{2} + 11 (- \frac{11}{2}) - 15$

Step 3.2.1.1.2

Apply the product rule to $\frac{11}{2}$ .

$f (- \frac{11}{2}) = {(- 1)}^{2} (\frac{11^{2}}{2^{2}}) + 11 (- \frac{11}{2}) - 15$

Step 3.2.1.2

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

$f (- \frac{11}{2}) = 1 (\frac{11^{2}}{2^{2}}) + 11 (- \frac{11}{2}) - 15$

Step 3.2.1.3

Multiply $\frac{11^{2}}{2^{2}}$ by $1$ .

$f (- \frac{11}{2}) = \frac{11^{2}}{2^{2}} + 11 (- \frac{11}{2}) - 15$

Step 3.2.1.4

Raise $11$ to the power of $2$ .

$f (- \frac{11}{2}) = \frac{121}{2^{2}} + 11 (- \frac{11}{2}) - 15$

Step 3.2.1.5

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

$f (- \frac{11}{2}) = \frac{121}{4} + 11 (- \frac{11}{2}) - 15$

Step 3.2.1.6

Multiply $11 (- \frac{11}{2})$ .

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Step 3.2.1.6.1

Multiply $- 1$ by $11$ .

$f (- \frac{11}{2}) = \frac{121}{4} - 11 (\frac{11}{2}) - 15$

Step 3.2.1.6.2

Combine $- 11$ and $\frac{11}{2}$ .

$f (- \frac{11}{2}) = \frac{121}{4} + \frac{- 11 \cdot 11}{2} - 15$

Step 3.2.1.6.3

Multiply $- 11$ by $11$ .

$f (- \frac{11}{2}) = \frac{121}{4} + \frac{- 121}{2} - 15$

Step 3.2.1.7

Move the negative in front of the fraction.

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121}{2} - 15$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply $\frac{121}{2}$ by $\frac{2}{2}$ .

$f (- \frac{11}{2}) = \frac{121}{4} - (\frac{121}{2} \cdot \frac{2}{2}) - 15$

Step 3.2.2.2

Multiply $\frac{121}{2}$ by $\frac{2}{2}$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{2 \cdot 2} - 15$

Step 3.2.2.3

Write $- 15$ as a fraction with denominator $1$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{2 \cdot 2} + \frac{- 15}{1}$

Step 3.2.2.4

Multiply $\frac{- 15}{1}$ by $\frac{4}{4}$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{2 \cdot 2} + \frac{- 15}{1} \cdot \frac{4}{4}$

Step 3.2.2.5

Multiply $\frac{- 15}{1}$ by $\frac{4}{4}$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{2 \cdot 2} + \frac{- 15 \cdot 4}{4}$

Step 3.2.2.6

Multiply $2$ by $2$ .

$f (- \frac{11}{2}) = \frac{121}{4} - \frac{121 \cdot 2}{4} + \frac{- 15 \cdot 4}{4}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{11}{2}) = \frac{121 - 121 \cdot 2 - 15 \cdot 4}{4}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $- 121$ by $2$ .

$f (- \frac{11}{2}) = \frac{121 - 242 - 15 \cdot 4}{4}$

Step 3.2.4.2

Multiply $- 15$ by $4$ .

$f (- \frac{11}{2}) = \frac{121 - 242 - 60}{4}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract $242$ from $121$ .

$f (- \frac{11}{2}) = \frac{- 121 - 60}{4}$

Step 3.2.5.2

Subtract $60$ from $- 121$ .

$f (- \frac{11}{2}) = \frac{- 181}{4}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- \frac{11}{2}) = - \frac{181}{4}$

Step 3.2.6

The final answer is $- \frac{181}{4}$ .

$- \frac{181}{4}$

Step 4

The horizontal tangent line on function $f (x) = x^{2} + 11 x - 15$ is $y = - \frac{181}{4}$ .

$y = - \frac{181}{4}$

Step 5