Calculus Examples

$f (x) = \frac{x^{2}}{x^{2} - 16}$

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 $g (x) = x^{2} - 16$ .

$f' (x) = \frac{(x^{2} - 16) \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} (x^{2} - 16)}{{(x^{2} - 16)}^{2}}$

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 $n = 2$ .

$f' (x) = \frac{(x^{2} - 16) (2 x) - x^{2} \frac{d}{d x} (x^{2} - 16)}{{(x^{2} - 16)}^{2}}$

Step 1.1.2.2

Move $2$ to the left of $x^{2} - 16$ .

$f' (x) = \frac{2 \cdot ((x^{2} - 16) x) - x^{2} \frac{d}{d x} (x^{2} - 16)}{{(x^{2} - 16)}^{2}}$

Step 1.1.2.3

By the Sum Rule, the derivative of $x^{2} - 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16]$ .

$f' (x) = \frac{2 (x^{2} - 16) x - x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 16))}{{(x^{2} - 16)}^{2}}$

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 $n = 2$ .

$f' (x) = \frac{2 (x^{2} - 16) x - x^{2} (2 x + \frac{d}{d x} (- 16))}{{(x^{2} - 16)}^{2}}$

Step 1.1.2.5

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x^{2} - 16) x - x^{2} (2 x + 0)}{{(x^{2} - 16)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{2 (x^{2} - 16) x - x^{2} (2 x)}{{(x^{2} - 16)}^{2}}$

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{2 (x^{2} - 16) x - 2 x^{2} x}{{(x^{2} - 16)}^{2}}$

Step 1.1.3

Raise $x$ to the power of $1$ .

$f' (x) = \frac{2 (x^{2} - 16) x - 2 (x \cdot x^{2})}{{(x^{2} - 16)}^{2}}$

Step 1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{2 (x^{2} - 16) x - 2 x^{1 + 2}}{{(x^{2} - 16)}^{2}}$

Step 1.1.5

Add $1$ and $2$ .

$f' (x) = \frac{2 (x^{2} - 16) x - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$f' (x) = \frac{(2 x^{2} + 2 \cdot - 16) x - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.2

Apply the distributive property.

$f' (x) = \frac{2 x^{2} x + 2 \cdot (- 16 x) - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3

Simplify the numerator.

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Step 1.1.6.3.1

Simplify each term.

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Step 1.1.6.3.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.6.3.1.1.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (- 16 x) - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.6.3.1.1.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (- 16 x) - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{2 x^{1 + 2} + 2 \cdot (- 16 x) - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3.1.1.3

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} + 2 \cdot (- 16 x) - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3.1.2

Multiply $2$ by $- 16$ .

$f' (x) = \frac{2 x^{3} - 32 x - 2 x^{3}}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3.2

Combine the opposite terms in $2 x^{3} - 32 x - 2 x^{3}$ .

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Step 1.1.6.3.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = \frac{- 32 x + 0}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.3.2.2

Add $- 32 x$ and $0$ .

$f' (x) = \frac{- 32 x}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.4

Move the negative in front of the fraction.

$f' (x) = - \frac{32 x}{{(x^{2} - 16)}^{2}}$

Step 1.1.6.5

Simplify the denominator.

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Step 1.1.6.5.1

Rewrite $16$ as $4^{2}$ .

$f' (x) = - \frac{32 x}{{(x^{2} - 4^{2})}^{2}}$

Step 1.1.6.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$f' (x) = - \frac{32 x}{{((x + 4) (x - 4))}^{2}}$

Step 1.1.6.5.3

Apply the product rule to $(x + 4) (x - 4)$ .

$f' (x) = - \frac{32 x}{{(x + 4)}^{2} {(x - 4)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{32 x}{{(x + 4)}^{2} {(x - 4)}^{2}}$ .

$- \frac{32 x}{{(x + 4)}^{2} {(x - 4)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{32 x}{{(x + 4)}^{2} {(x - 4)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{32 x}{{(x + 4)}^{2} {(x - 4)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$32 x = 0$

Step 2.3

Divide each term in $32 x = 0$ by $32$ and simplify.

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Step 2.3.1

Divide each term in $32 x = 0$ by $32$ .

$\frac{32 x}{32} = \frac{0}{32}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $32$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{32 x}{32} = \frac{0}{32}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{32}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $32$ .

$x = 0$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{32 x}{{(x + 4)}^{2} {(x - 4)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 4)}^{2} {(x - 4)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

${(x + 4)}^{2} = 0$

${(x - 4)}^{2} = 0$

Step 3.2.2

Set ${(x + 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.2.1

Set ${(x + 4)}^{2}$ equal to $0$ .

${(x + 4)}^{2} = 0$

Step 3.2.2.2

Solve ${(x + 4)}^{2} = 0$ for $x$ .

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Step 3.2.2.2.1

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 3.2.2.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.2.3

Set ${(x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x - 4)}^{2}$ equal to $0$ .

${(x - 4)}^{2} = 0$

Step 3.2.3.2

Solve ${(x - 4)}^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.2.3.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.2.4

The final solution is all the values that make ${(x + 4)}^{2} {(x - 4)}^{2} = 0$ true.

$x = - 4, 4$

Step 3.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 4, x = 4$

Step 4

Evaluate $\frac{x^{2}}{x^{2} - 16}$ 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 $x$ .

$\frac{{(0)}^{2}}{{(0)}^{2} - 16}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\frac{0}{0^{2} - 16}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

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

$\frac{0}{0 - 16}$

Step 4.1.2.2.2

Subtract $16$ from $0$ .

$\frac{0}{- 16}$

Step 4.1.2.3

Divide $0$ by $- 16$ .

$0$

Step 4.2

Evaluate at $x = - 4$ .

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Step 4.2.1

Substitute $- 4$ for $x$ .

$\frac{{(- 4)}^{2}}{{(- 4)}^{2} - 16}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Raise $- 4$ to the power of $2$ .

$\frac{{(- 4)}^{2}}{16 - 16}$

Step 4.2.2.2

Subtract $16$ from $16$ .

$\frac{{(- 4)}^{2}}{0}$

Step 4.2.2.3

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

Undefined

Step 4.3

Evaluate at $x = 4$ .

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Step 4.3.1

Substitute $4$ for $x$ .

$\frac{{(4)}^{2}}{{(4)}^{2} - 16}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Raise $4$ to the power of $2$ .

$\frac{{(4)}^{2}}{16 - 16}$

Step 4.3.2.2

Subtract $16$ from $16$ .

$\frac{{(4)}^{2}}{0}$

Step 4.3.2.3

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

Undefined

Step 4.4

List all of the points.

$(0, 0)$

Step 5