Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2}$ and $g (x) = x - 9$ .

$\frac{(x - 9) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x - 9]}{{(x - 9)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{(x - 9) (2 x) - x^{2} \frac{d}{d x} [x - 9]}{{(x - 9)}^{2}}$

Step 1.2.2

Move $2$ to the left of $x - 9$ .

$\frac{2 \cdot (x - 9) x - x^{2} \frac{d}{d x} [x - 9]}{{(x - 9)}^{2}}$

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{2 (x - 9) x - x^{2} (1 + 0)}{{(x - 9)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $1$ and $0$ .

$\frac{2 (x - 9) x - x^{2} \cdot 1}{{(x - 9)}^{2}}$

Step 1.2.6.2

Multiply $- 1$ by $1$ .

$\frac{2 (x - 9) x - x^{2}}{{(x - 9)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot - 9) x - x^{2}}{{(x - 9)}^{2}}$

Step 1.3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 9 x - x^{2}}{{(x - 9)}^{2}}$

Step 1.3.3

Simplify the numerator.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.3.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 \cdot - 9 x - x^{2}}{{(x - 9)}^{2}}$

Step 1.3.3.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 \cdot - 9 x - x^{2}}{{(x - 9)}^{2}}$

Step 1.3.3.1.2

Multiply $2$ by $- 9$ .

$\frac{2 x^{2} - 18 x - x^{2}}{{(x - 9)}^{2}}$

Step 1.3.3.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} - 18 x}{{(x - 9)}^{2}}$

Step 1.3.4

Factor $x$ out of $x^{2} - 18 x$ .

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Step 1.3.4.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x - 18 x}{{(x - 9)}^{2}}$

Step 1.3.4.2

Factor $x$ out of $- 18 x$ .

$\frac{x \cdot x + x \cdot - 18}{{(x - 9)}^{2}}$

Step 1.3.4.3

Factor $x$ out of $x \cdot x + x \cdot - 18$ .

$\frac{x (x - 18)}{{(x - 9)}^{2}}$

Step 2

Set the derivative equal to $0$ then solve the equation $\frac{x (x - 18)}{{(x - 9)}^{2}} = 0$ .

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Step 2.1

Set the numerator equal to zero.

$x (x - 18) = 0$

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 18 = 0$

Step 2.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3

Set $x - 18$ equal to $0$ and solve for $x$ .

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Step 2.2.3.1

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Step 2.2.3.2

Add $18$ to both sides of the equation.

$x = 18$

Step 2.2.4

The final solution is all the values that make $x (x - 18) = 0$ true.

$x = 0, 18$

Step 3

Solve the original function $f (x) = \frac{x^{2}}{x - 9}$ at $x = 0$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{2}}{(0) - 9}$

Step 3.2

Simplify the result.

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Step 3.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{0}{0 - 9}$

Step 3.2.2

Subtract $9$ from $0$ .

$f (0) = \frac{0}{- 9}$

Step 3.2.3

Divide $0$ by $- 9$ .

$f (0) = 0$

Step 3.2.4

The final answer is $0$ .

$0$

Step 4

Solve the original function $f (x) = \frac{x^{2}}{x - 9}$ at $x = 18$ .

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Step 4.1

Replace the variable $x$ with $18$ in the expression.

$f (18) = \frac{{(18)}^{2}}{(18) - 9}$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $18$ to the power of $2$ .

$f (18) = \frac{324}{18 - 9}$

Step 4.2.2

Subtract $9$ from $18$ .

$f (18) = \frac{324}{9}$

Step 4.2.3

Divide $324$ by $9$ .

$f (18) = 36$

Step 4.2.4

The final answer is $36$ .

$36$

Step 5

The horizontal tangent lines on function $f (x) = \frac{x^{2}}{x - 9}$ are $y = 0, y = 36$ .

$y = 0, y = 36$

Step 6