Calculus Examples

$y = \frac{1}{2} x^{2} + 9 x$

Step 1

Combine $\frac{1}{2}$ and $x^{2}$ .

$y = \frac{x^{2}}{2} + 9 x$

Step 2

Set $y$ as a function of $x$ .

$f (x) = \frac{x^{2}}{2} + 9 x$

Step 3

Find the derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{x^{2}}{2} + 9 x$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [9 x]$ .

$\frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [9 x]$

Step 3.2

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

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Step 3.2.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 x]$

Step 3.2.2

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

$\frac{1}{2} (2 x) + \frac{d}{d x} [9 x]$

Step 3.2.3

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} x + \frac{d}{d x} [9 x]$

Step 3.2.4

Combine $\frac{2}{2}$ and $x$ .

$\frac{2 x}{2} + \frac{d}{d x} [9 x]$

Step 3.2.5

Cancel the common factor of $2$ .

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Step 3.2.5.1

Cancel the common factor.

$\frac{2 x}{2} + \frac{d}{d x} [9 x]$

Step 3.2.5.2

Divide $x$ by $1$ .

$x + \frac{d}{d x} [9 x]$

Step 3.3

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 3.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$x + 9 \frac{d}{d x} [x]$

Step 3.3.2

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

$x + 9 \cdot 1$

Step 3.3.3

Multiply $9$ by $1$ .

$x + 9$

Step 4

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 5

Solve the original function $f (x) = \frac{x^{2}}{2} + 9 x$ at $x = - 9$ .

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Step 5.1

Replace the variable $x$ with $- 9$ in the expression.

$f (- 9) = \frac{{(- 9)}^{2}}{2} + 9 (- 9)$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 9$ to the power of $2$ .

$f (- 9) = \frac{81}{2} + 9 (- 9)$

Step 5.2.1.2

Multiply $9$ by $- 9$ .

$f (- 9) = \frac{81}{2} - 81$

Step 5.2.2

To write $- 81$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 9) = \frac{81}{2} - 81 \cdot \frac{2}{2}$

Step 5.2.3

Combine $- 81$ and $\frac{2}{2}$ .

$f (- 9) = \frac{81}{2} + \frac{- 81 \cdot 2}{2}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (- 9) = \frac{81 - 81 \cdot 2}{2}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $- 81$ by $2$ .

$f (- 9) = \frac{81 - 162}{2}$

Step 5.2.5.2

Subtract $162$ from $81$ .

$f (- 9) = \frac{- 81}{2}$

Step 5.2.6

Move the negative in front of the fraction.

$f (- 9) = - \frac{81}{2}$

Step 5.2.7

The final answer is $- \frac{81}{2}$ .

$- \frac{81}{2}$

Step 6

The horizontal tangent line on function $f (x) = \frac{x^{2}}{2} + 9 x$ is $y = - \frac{81}{2}$ .

$y = - \frac{81}{2}$

Step 7