Calculus Examples

$y = \frac{x^{2}}{x - 6}$

Step 1

Write $y = \frac{x^{2}}{x - 6}$ as a function.

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

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

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 $g (x) = x - 6$ .

$\frac{(x - 6) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

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

$\frac{(x - 6) (2 x) - x^{2} \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.1.1.2.2

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

$\frac{2 \cdot (x - 6) x - x^{2} \frac{d}{d x} [x - 6]}{{(x - 6)}^{2}}$

Step 2.1.1.2.3

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

$\frac{2 (x - 6) x - x^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 6])}{{(x - 6)}^{2}}$

Step 2.1.1.2.4

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

$\frac{2 (x - 6) x - x^{2} (1 + \frac{d}{d x} [- 6])}{{(x - 6)}^{2}}$

Step 2.1.1.2.5

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

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

Step 2.1.1.2.6

Simplify the expression.

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Step 2.1.1.2.6.1

Add $1$ and $0$ .

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

Step 2.1.1.2.6.2

Multiply $- 1$ by $1$ .

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

Step 2.1.1.3

Simplify.

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Step 2.1.1.3.1

Apply the distributive property.

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

Step 2.1.1.3.2

Apply the distributive property.

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

Step 2.1.1.3.3

Simplify the numerator.

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Step 2.1.1.3.3.1

Simplify each term.

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Step 2.1.1.3.3.1.1

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

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Step 2.1.1.3.3.1.1.1

Move $x$ .

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

Step 2.1.1.3.3.1.1.2

Multiply $x$ by $x$ .

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

Step 2.1.1.3.3.1.2

Multiply $2$ by $- 6$ .

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

Step 2.1.1.3.3.2

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

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

Step 2.1.1.3.4

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

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Step 2.1.1.3.4.1

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

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

Step 2.1.1.3.4.2

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

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

Step 2.1.1.3.4.3

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

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

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

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

Step 2.1.2.2

Multiply the exponents in ${({(x - 6)}^{2})}^{2}$ .

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Step 2.1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(x - 6)}^{2} \frac{d}{d x} [x (x - 12)] - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{2 \cdot 2}}$

Step 2.1.2.2.2

Multiply $2$ by $2$ .

$\frac{{(x - 6)}^{2} \frac{d}{d x} [x (x - 12)] - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 12$ .

$\frac{{(x - 6)}^{2} (x \frac{d}{d x} [x - 12] + (x - 12) \frac{d}{d x} [x]) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4

Differentiate.

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Step 2.1.2.4.1

By the Sum Rule, the derivative of $x - 12$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 12]$ .

$\frac{{(x - 6)}^{2} (x (\frac{d}{d x} [x] + \frac{d}{d x} [- 12]) + (x - 12) \frac{d}{d x} [x]) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.2

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

$\frac{{(x - 6)}^{2} (x (1 + \frac{d}{d x} [- 12]) + (x - 12) \frac{d}{d x} [x]) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.3

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{{(x - 6)}^{2} (x (1 + 0) + (x - 12) \frac{d}{d x} [x]) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.4

Simplify the expression.

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Step 2.1.2.4.4.1

Add $1$ and $0$ .

$\frac{{(x - 6)}^{2} (x \cdot 1 + (x - 12) \frac{d}{d x} [x]) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.4.2

Multiply $x$ by $1$ .

$\frac{{(x - 6)}^{2} (x + (x - 12) \frac{d}{d x} [x]) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.5

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

$\frac{{(x - 6)}^{2} (x + (x - 12) \cdot 1) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.6

Simplify by adding terms.

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Step 2.1.2.4.6.1

Multiply $x - 12$ by $1$ .

$\frac{{(x - 6)}^{2} (x + x - 12) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.4.6.2

Add $x$ and $x$ .

$\frac{{(x - 6)}^{2} (2 x - 12) - x (x - 12) \frac{d}{d x} [{(x - 6)}^{2}]}{{(x - 6)}^{4}}$

Step 2.1.2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 6$ .

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Step 2.1.2.5.1

To apply the Chain Rule, set $u$ as $x - 6$ .

$\frac{{(x - 6)}^{2} (2 x - 12) - x (x - 12) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 6])}{{(x - 6)}^{4}}$

Step 2.1.2.5.2

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

$\frac{{(x - 6)}^{2} (2 x - 12) - x (x - 12) (2 u \frac{d}{d x} [x - 6])}{{(x - 6)}^{4}}$

Step 2.1.2.5.3

Replace all occurrences of $u$ with $x - 6$ .

$\frac{{(x - 6)}^{2} (2 x - 12) - x (x - 12) (2 (x - 6) \frac{d}{d x} [x - 6])}{{(x - 6)}^{4}}$

Step 2.1.2.6

Simplify with factoring out.

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Step 2.1.2.6.1

Multiply $2$ by $- 1$ .

$\frac{{(x - 6)}^{2} (2 x - 12) - 2 x (x - 12) ((x - 6) \frac{d}{d x} [x - 6])}{{(x - 6)}^{4}}$

Step 2.1.2.6.2

Factor $x - 6$ out of ${(x - 6)}^{2} (2 x - 12) - 2 x (x - 12) ((x - 6) \frac{d}{d x} [x - 6])$ .

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Step 2.1.2.6.2.1

Factor $x - 6$ out of ${(x - 6)}^{2} (2 x - 12)$ .

$\frac{(x - 6) ((x - 6) (2 x - 12)) - 2 x (x - 12) ((x - 6) \frac{d}{d x} [x - 6])}{{(x - 6)}^{4}}$

Step 2.1.2.6.2.2

Factor $x - 6$ out of $- 2 x (x - 12) ((x - 6) \frac{d}{d x} [x - 6])$ .

$\frac{(x - 6) ((x - 6) (2 x - 12)) + (x - 6) (- 2 x (x - 12) (\frac{d}{d x} [x - 6]))}{{(x - 6)}^{4}}$

Step 2.1.2.6.2.3

Factor $x - 6$ out of $(x - 6) ((x - 6) (2 x - 12)) + (x - 6) (- 2 x (x - 12) (\frac{d}{d x} [x - 6]))$ .

$\frac{(x - 6) ((x - 6) (2 x - 12) - 2 x (x - 12) (\frac{d}{d x} [x - 6]))}{{(x - 6)}^{4}}$

Step 2.1.2.7

Cancel the common factors.

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Step 2.1.2.7.1

Factor $x - 6$ out of ${(x - 6)}^{4}$ .

$\frac{(x - 6) ((x - 6) (2 x - 12) - 2 x (x - 12) (\frac{d}{d x} [x - 6]))}{(x - 6) {(x - 6)}^{3}}$

Step 2.1.2.7.2

Cancel the common factor.

$\frac{(x - 6) ((x - 6) (2 x - 12) - 2 x (x - 12) (\frac{d}{d x} [x - 6]))}{(x - 6) {(x - 6)}^{3}}$

Step 2.1.2.7.3

Rewrite the expression.

$\frac{(x - 6) (2 x - 12) - 2 x (x - 12) (\frac{d}{d x} [x - 6])}{{(x - 6)}^{3}}$

Step 2.1.2.8

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

$\frac{(x - 6) (2 x - 12) - 2 x (x - 12) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6])}{{(x - 6)}^{3}}$

Step 2.1.2.9

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

$\frac{(x - 6) (2 x - 12) - 2 x (x - 12) (1 + \frac{d}{d x} [- 6])}{{(x - 6)}^{3}}$

Step 2.1.2.10

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

$\frac{(x - 6) (2 x - 12) - 2 x (x - 12) (1 + 0)}{{(x - 6)}^{3}}$

Step 2.1.2.11

Simplify the expression.

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Step 2.1.2.11.1

Add $1$ and $0$ .

$\frac{(x - 6) (2 x - 12) - 2 x (x - 12) \cdot 1}{{(x - 6)}^{3}}$

Step 2.1.2.11.2

Multiply $- 2$ by $1$ .

$\frac{(x - 6) (2 x - 12) - 2 x (x - 12)}{{(x - 6)}^{3}}$

Step 2.1.2.12

Simplify.

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Step 2.1.2.12.1

Apply the distributive property.

$\frac{(x - 6) (2 x - 12) - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2

Simplify the numerator.

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Step 2.1.2.12.2.1

Simplify each term.

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Step 2.1.2.12.2.1.1

Expand $(x - 6) (2 x - 12)$ using the FOIL Method.

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Step 2.1.2.12.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 12) - 6 (2 x - 12) - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 12 - 6 (2 x - 12) - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 12 - 6 (2 x) - 6 \cdot - 12 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2

Simplify and combine like terms.

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Step 2.1.2.12.2.1.2.1

Simplify each term.

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Step 2.1.2.12.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 12 - 6 (2 x) - 6 \cdot - 12 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2.1.2

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

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Step 2.1.2.12.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 12 - 6 (2 x) - 6 \cdot - 12 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 12 - 6 (2 x) - 6 \cdot - 12 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2.1.3

Move $- 12$ to the left of $x$ .

$\frac{2 x^{2} - 12 \cdot x - 6 (2 x) - 6 \cdot - 12 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2.1.4

Multiply $2$ by $- 6$ .

$\frac{2 x^{2} - 12 x - 12 x - 6 \cdot - 12 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2.1.5

Multiply $- 6$ by $- 12$ .

$\frac{2 x^{2} - 12 x - 12 x + 72 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.2.2

Subtract $12 x$ from $- 12 x$ .

$\frac{2 x^{2} - 24 x + 72 - 2 x \cdot x - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.3

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

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Step 2.1.2.12.2.1.3.1

Move $x$ .

$\frac{2 x^{2} - 24 x + 72 - 2 (x \cdot x) - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.3.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 24 x + 72 - 2 x^{2} - 2 x \cdot - 12}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.1.4

Multiply $- 12$ by $- 2$ .

$\frac{2 x^{2} - 24 x + 72 - 2 x^{2} + 24 x}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.2

Combine the opposite terms in $2 x^{2} - 24 x + 72 - 2 x^{2} + 24 x$ .

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Step 2.1.2.12.2.2.1

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

$\frac{- 24 x + 72 + 0 + 24 x}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.2.2

Add $- 24 x + 72$ and $0$ .

$\frac{- 24 x + 72 + 24 x}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.2.3

Add $- 24 x$ and $24 x$ .

$\frac{0 + 72}{{(x - 6)}^{3}}$

Step 2.1.2.12.2.2.4

Add $0$ and $72$ .

$f'' (x) = \frac{72}{{(x - 6)}^{3}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{72}{{(x - 6)}^{3}}$ .

$\frac{72}{{(x - 6)}^{3}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{72}{{(x - 6)}^{3}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{72}{{(x - 6)}^{3}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$72 = 0$

Step 2.2.3

Since $72 \neq 0$ , there are no solutions.

No solution

Step 3

Find the domain of $f (x) = \frac{x^{2}}{x - 6}$ .

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Step 3.1

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

$x - 6 = 0$

Step 3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq 6}$

Interval Notation:

$(- \infty, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq 6}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 6) \cup (6, \infty)$

Step 5

Substitute any number from the interval $(- \infty, 6)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (0) = \frac{72}{{((0) - 6)}^{3}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the denominator.

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Step 5.2.1.1

Subtract $6$ from $0$ .

$f'' (0) = \frac{72}{{(- 6)}^{3}}$

Step 5.2.1.2

Raise $- 6$ to the power of $3$ .

$f'' (0) = \frac{72}{- 216}$

Step 5.2.2

Reduce the expression by cancelling the common factors.

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Step 5.2.2.1

Cancel the common factor of $72$ and $- 216$ .

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Step 5.2.2.1.1

Factor $72$ out of $72$ .

$f'' (0) = \frac{72 (1)}{- 216}$

Step 5.2.2.1.2

Cancel the common factors.

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Step 5.2.2.1.2.1

Factor $72$ out of $- 216$ .

$f'' (0) = \frac{72 \cdot 1}{72 \cdot - 3}$

Step 5.2.2.1.2.2

Cancel the common factor.

$f'' (0) = \frac{72 \cdot 1}{72 \cdot - 3}$

Step 5.2.2.1.2.3

Rewrite the expression.

$f'' (0) = \frac{1}{- 3}$

Step 5.2.2.2

Move the negative in front of the fraction.

$f'' (0) = - \frac{1}{3}$

Step 5.2.3

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 5.3

The graph is concave down on the interval $(- \infty, 6)$ because $f'' (0)$ is negative.

Concave down on $(- \infty, 6)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(6, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $8$ in the expression.

$f'' (8) = \frac{72}{{((8) - 6)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the denominator.

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Step 6.2.1.1

Subtract $6$ from $8$ .

$f'' (8) = \frac{72}{2^{3}}$

Step 6.2.1.2

Raise $2$ to the power of $3$ .

$f'' (8) = \frac{72}{8}$

Step 6.2.2

Divide $72$ by $8$ .

$f'' (8) = 9$

Step 6.2.3

The final answer is $9$ .

$9$

Step 6.3

The graph is concave up on the interval $(6, \infty)$ because $f'' (8)$ is positive.

Concave up on $(6, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, 6)$ since $f'' (x)$ is negative

Concave up on $(6, \infty)$ since $f'' (x)$ is positive

Step 8