Calculus Examples

$f (x) = \frac{x^{2}}{x - 6}$

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 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 $g (x) = x - 6$ .

$f' (x) = \frac{(x - 6) \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} (x - 6)}{{(x - 6)}^{2}}$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

$f' (x) = \frac{(x - 6) (2 x) - x^{2} \frac{d}{d x} (x - 6)}{{(x - 6)}^{2}}$

Step 1.1.2.2

Move $2$ to the left of $x - 6$ .

$f' (x) = \frac{2 \cdot ((x - 6) x) - x^{2} \frac{d}{d x} (x - 6)}{{(x - 6)}^{2}}$

Step 1.1.2.3

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$f' (x) = \frac{2 (x - 6) x - x^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (- 6))}{{(x - 6)}^{2}}$

Step 1.1.2.4

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

$f' (x) = \frac{2 (x - 6) x - x^{2} (1 + \frac{d}{d x} (- 6))}{{(x - 6)}^{2}}$

Step 1.1.2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x - 6) x - x^{2} (1 + 0)}{{(x - 6)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $1$ and $0$ .

$f' (x) = \frac{2 (x - 6) x - x^{2} \cdot 1}{{(x - 6)}^{2}}$

Step 1.1.2.6.2

Multiply $- 1$ by $1$ .

$f' (x) = \frac{2 (x - 6) x - x^{2}}{{(x - 6)}^{2}}$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$f' (x) = \frac{(2 x + 2 \cdot - 6) x - x^{2}}{{(x - 6)}^{2}}$

Step 1.1.3.2

Apply the distributive property.

$f' (x) = \frac{2 x \cdot x + 2 \cdot (- 6 x) - x^{2}}{{(x - 6)}^{2}}$

Step 1.1.3.3

Simplify the numerator.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

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Step 1.1.3.3.1.1.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x) + 2 \cdot (- 6 x) - x^{2}}{{(x - 6)}^{2}}$

Step 1.1.3.3.1.1.2

Multiply $x$ by $x$ .

$f' (x) = \frac{2 x^{2} + 2 \cdot (- 6 x) - x^{2}}{{(x - 6)}^{2}}$

Step 1.1.3.3.1.2

Multiply $2$ by $- 6$ .

$f' (x) = \frac{2 x^{2} - 12 x - x^{2}}{{(x - 6)}^{2}}$

Step 1.1.3.3.2

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

$f' (x) = \frac{x^{2} - 12 x}{{(x - 6)}^{2}}$

Step 1.1.3.4

Factor $x$ out of $x^{2} - 12 x$ .

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Step 1.1.3.4.1

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

$f' (x) = \frac{x \cdot x - 12 x}{{(x - 6)}^{2}}$

Step 1.1.3.4.2

Factor $x$ out of $- 12 x$ .

$f' (x) = \frac{x \cdot x + x \cdot - 12}{{(x - 6)}^{2}}$

Step 1.1.3.4.3

Factor $x$ out of $x \cdot x + x \cdot - 12$ .

$f' (x) = \frac{x (x - 12)}{{(x - 6)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x (x - 12)}{{(x - 6)}^{2}}$ .

$\frac{x (x - 12)}{{(x - 6)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x (x - 12)}{{(x - 6)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x (x - 12)}{{(x - 6)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$x (x - 12) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x = 0$

$x - 12 = 0$

Step 2.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3.3

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 2.3.3.2

Add $12$ to both sides of the equation.

$x = 12$

Step 2.3.4

The final solution is all the values that make $x (x - 12) = 0$ true.

$x = 0, 12$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x (x - 12)}{{(x - 6)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 6)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Evaluate $\frac{x^{2}}{x - 6}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{{(0)}^{2}}{(0) - 6}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\frac{0}{0 - 6}$

Step 4.1.2.2

Subtract $6$ from $0$ .

$\frac{0}{- 6}$

Step 4.1.2.3

Divide $0$ by $- 6$ .

$0$

Step 4.2

Evaluate at $x = 12$ .

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Step 4.2.1

Substitute $12$ for $x$ .

$\frac{{(12)}^{2}}{(12) - 6}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $12$ to the power of $2$ .

$\frac{144}{12 - 6}$

Step 4.2.2.2

Subtract $6$ from $12$ .

$\frac{144}{6}$

Step 4.2.2.3

Divide $144$ by $6$ .

$24$

Step 4.3

Evaluate at $x = 6$ .

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Step 4.3.1

Substitute $6$ for $x$ .

$\frac{{(6)}^{2}}{(6) - 6}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract $6$ from $6$ .

$\frac{{(6)}^{2}}{0}$

Step 4.3.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.4

List all of the points.

$(0, 0), (12, 24)$

Step 5