Calculus Examples

$y = \frac{x^{2}}{x^{2} + 243}$

Step 1

Write $y = \frac{x^{2}}{x^{2} + 243}$ as a function.

$f (x) = \frac{x^{2}}{x^{2} + 243}$

Step 2

Find the first derivative of the function.

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Step 2.1

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 $g (x) = x^{2} + 243$ .

$\frac{(x^{2} + 243) \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [x^{2} + 243]}{{(x^{2} + 243)}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

$\frac{(x^{2} + 243) (2 x) - x^{2} \frac{d}{d x} [x^{2} + 243]}{{(x^{2} + 243)}^{2}}$

Step 2.2.2

Move $2$ to the left of $x^{2} + 243$ .

$\frac{2 \cdot (x^{2} + 243) x - x^{2} \frac{d}{d x} [x^{2} + 243]}{{(x^{2} + 243)}^{2}}$

Step 2.2.3

By the Sum Rule, the derivative of $x^{2} + 243$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [243]$ .

$\frac{2 (x^{2} + 243) x - x^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [243])}{{(x^{2} + 243)}^{2}}$

Step 2.2.4

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

$\frac{2 (x^{2} + 243) x - x^{2} (2 x + \frac{d}{d x} [243])}{{(x^{2} + 243)}^{2}}$

Step 2.2.5

Since $243$ is constant with respect to $x$ , the derivative of $243$ with respect to $x$ is $0$ .

$\frac{2 (x^{2} + 243) x - x^{2} (2 x + 0)}{{(x^{2} + 243)}^{2}}$

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Add $2 x$ and $0$ .

$\frac{2 (x^{2} + 243) x - x^{2} (2 x)}{{(x^{2} + 243)}^{2}}$

Step 2.2.6.2

Multiply $2$ by $- 1$ .

$\frac{2 (x^{2} + 243) x - 2 x^{2} x}{{(x^{2} + 243)}^{2}}$

Step 2.3

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

$\frac{2 (x^{2} + 243) x - 2 (x^{1} x^{2})}{{(x^{2} + 243)}^{2}}$

Step 2.4

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

$\frac{2 (x^{2} + 243) x - 2 x^{1 + 2}}{{(x^{2} + 243)}^{2}}$

Step 2.5

Add $1$ and $2$ .

$\frac{2 (x^{2} + 243) x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

$\frac{(2 x^{2} + 2 \cdot 243) x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.2

Apply the distributive property.

$\frac{2 x^{2} x + 2 \cdot 243 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.3

Simplify the numerator.

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Step 2.6.3.1

Simplify each term.

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Step 2.6.3.1.1

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

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Step 2.6.3.1.1.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + 2 \cdot 243 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.3.1.1.2

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

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Step 2.6.3.1.1.2.1

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

$\frac{2 (x^{1} x^{2}) + 2 \cdot 243 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.3.1.1.2.2

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

$\frac{2 x^{1 + 2} + 2 \cdot 243 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.3.1.1.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 2 \cdot 243 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.3.1.2

Multiply $2$ by $243$ .

$\frac{2 x^{3} + 486 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 2.6.3.2

Combine the opposite terms in $2 x^{3} + 486 x - 2 x^{3}$ .

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Step 2.6.3.2.1

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

$\frac{486 x + 0}{{(x^{2} + 243)}^{2}}$

Step 2.6.3.2.2

Add $486 x$ and $0$ .

$\frac{486 x}{{(x^{2} + 243)}^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $486$ is constant with respect to $x$ , the derivative of $\frac{486 x}{{(x^{2} + 243)}^{2}}$ with respect to $x$ is $486 \frac{d}{d x} [\frac{x}{{(x^{2} + 243)}^{2}}]$ .

$f'' (x) = 486 \frac{d}{d x} (\frac{x}{{(x^{2} + 243)}^{2}})$

Step 3.2

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 243)}^{2}}{{({(x^{2} + 243)}^{2})}^{2}})$

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

Multiply the exponents in ${({(x^{2} + 243)}^{2})}^{2}$ .

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Step 3.3.1.1

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

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 243)}^{2}}{{(x^{2} + 243)}^{2 \cdot 2}})$

Step 3.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} \frac{d}{d x} (x) - x \frac{d}{d x} {(x^{2} + 243)}^{2}}{{(x^{2} + 243)}^{4}})$

Step 3.3.2

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

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} \cdot 1 - x \frac{d}{d x} {(x^{2} + 243)}^{2}}{{(x^{2} + 243)}^{4}})$

Step 3.3.3

Multiply ${(x^{2} + 243)}^{2}$ by $1$ .

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} - x \frac{d}{d x} {(x^{2} + 243)}^{2}}{{(x^{2} + 243)}^{4}})$

Step 3.4

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 $g (x) = x^{2} + 243$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{2} + 243$ .

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} - x (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 243))}{{(x^{2} + 243)}^{4}})$

Step 3.4.2

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

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} - x (2 u \frac{d}{d x} (x^{2} + 243))}{{(x^{2} + 243)}^{4}})$

Step 3.4.3

Replace all occurrences of $u$ with $x^{2} + 243$ .

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} - x (2 (x^{2} + 243) \frac{d}{d x} (x^{2} + 243))}{{(x^{2} + 243)}^{4}})$

Step 3.5

Simplify with factoring out.

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Step 3.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = 486 (\frac{{(x^{2} + 243)}^{2} - 2 x ((x^{2} + 243) \frac{d}{d x} (x^{2} + 243))}{{(x^{2} + 243)}^{4}})$

Step 3.5.2

Factor $x^{2} + 243$ out of ${(x^{2} + 243)}^{2} - 2 x ((x^{2} + 243) \frac{d}{d x} [x^{2} + 243])$ .

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Step 3.5.2.1

Factor $x^{2} + 243$ out of ${(x^{2} + 243)}^{2}$ .

$f'' (x) = 486 (\frac{(x^{2} + 243) (x^{2} + 243) - 2 x ((x^{2} + 243) \frac{d}{d x} (x^{2} + 243))}{{(x^{2} + 243)}^{4}})$

Step 3.5.2.2

Factor $x^{2} + 243$ out of $- 2 x ((x^{2} + 243) \frac{d}{d x} [x^{2} + 243])$ .

$f'' (x) = 486 (\frac{(x^{2} + 243) (x^{2} + 243) + (x^{2} + 243) (- 2 x (\frac{d}{d x} (x^{2} + 243)))}{{(x^{2} + 243)}^{4}})$

Step 3.5.2.3

Factor $x^{2} + 243$ out of $(x^{2} + 243) (x^{2} + 243) + (x^{2} + 243) (- 2 x (\frac{d}{d x} [x^{2} + 243]))$ .

$f'' (x) = 486 (\frac{(x^{2} + 243) (x^{2} + 243 - 2 x (\frac{d}{d x} (x^{2} + 243)))}{{(x^{2} + 243)}^{4}})$

Step 3.6

Cancel the common factors.

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Step 3.6.1

Factor $x^{2} + 243$ out of ${(x^{2} + 243)}^{4}$ .

$f'' (x) = 486 (\frac{(x^{2} + 243) (x^{2} + 243 - 2 x (\frac{d}{d x} (x^{2} + 243)))}{(x^{2} + 243) {(x^{2} + 243)}^{3}})$

Step 3.6.2

Cancel the common factor.

$f'' (x) = 486 (\frac{(x^{2} + 243) (x^{2} + 243 - 2 x (\frac{d}{d x} (x^{2} + 243)))}{(x^{2} + 243) {(x^{2} + 243)}^{3}})$

Step 3.6.3

Rewrite the expression.

$f'' (x) = 486 (\frac{x^{2} + 243 - 2 x (\frac{d}{d x} (x^{2} + 243))}{{(x^{2} + 243)}^{3}})$

Step 3.7

By the Sum Rule, the derivative of $x^{2} + 243$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [243]$ .

$f'' (x) = 486 (\frac{x^{2} + 243 - 2 x (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (243))}{{(x^{2} + 243)}^{3}})$

Step 3.8

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

$f'' (x) = 486 (\frac{x^{2} + 243 - 2 x (2 x + \frac{d}{d x} (243))}{{(x^{2} + 243)}^{3}})$

Step 3.9

Since $243$ is constant with respect to $x$ , the derivative of $243$ with respect to $x$ is $0$ .

$f'' (x) = 486 (\frac{x^{2} + 243 - 2 x (2 x + 0)}{{(x^{2} + 243)}^{3}})$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $2 x$ and $0$ .

$f'' (x) = 486 (\frac{x^{2} + 243 - 2 x (2 x)}{{(x^{2} + 243)}^{3}})$

Step 3.10.2

Multiply $2$ by $- 2$ .

$f'' (x) = 486 (\frac{x^{2} + 243 - 4 x \cdot x}{{(x^{2} + 243)}^{3}})$

Step 3.11

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

$f'' (x) = 486 (\frac{x^{2} + 243 - 4 (x \cdot x)}{{(x^{2} + 243)}^{3}})$

Step 3.12

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

$f'' (x) = 486 (\frac{x^{2} + 243 - 4 (x \cdot x)}{{(x^{2} + 243)}^{3}})$

Step 3.13

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

$f'' (x) = 486 (\frac{x^{2} + 243 - 4 x^{1 + 1}}{{(x^{2} + 243)}^{3}})$

Step 3.14

Add $1$ and $1$ .

$f'' (x) = 486 (\frac{x^{2} + 243 - 4 x^{2}}{{(x^{2} + 243)}^{3}})$

Step 3.15

Subtract $4 x^{2}$ from $x^{2}$ .

$f'' (x) = 486 (\frac{- 3 x^{2} + 243}{{(x^{2} + 243)}^{3}})$

Step 3.16

Combine $486$ and $\frac{- 3 x^{2} + 243}{{(x^{2} + 243)}^{3}}$ .

$f'' (x) = \frac{486 (- 3 x^{2} + 243)}{{(x^{2} + 243)}^{3}}$

Step 3.17

Simplify.

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Step 3.17.1

Apply the distributive property.

$f'' (x) = \frac{486 (- 3 x^{2}) + 486 \cdot 243}{{(x^{2} + 243)}^{3}}$

Step 3.17.2

Simplify each term.

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Step 3.17.2.1

Multiply $- 3$ by $486$ .

$f'' (x) = \frac{- 1458 x^{2} + 486 \cdot 243}{{(x^{2} + 243)}^{3}}$

Step 3.17.2.2

Multiply $486$ by $243$ .

$f'' (x) = \frac{- 1458 x^{2} + 118098}{{(x^{2} + 243)}^{3}}$

Step 3.17.3

Simplify the numerator.

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Step 3.17.3.1

Factor $1458$ out of $- 1458 x^{2} + 118098$ .

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Step 3.17.3.1.1

Factor $1458$ out of $- 1458 x^{2}$ .

$f'' (x) = \frac{1458 (- x^{2}) + 118098}{{(x^{2} + 243)}^{3}}$

Step 3.17.3.1.2

Factor $1458$ out of $118098$ .

$f'' (x) = \frac{1458 (- x^{2}) + 1458 (81)}{{(x^{2} + 243)}^{3}}$

Step 3.17.3.1.3

Factor $1458$ out of $1458 (- x^{2}) + 1458 (81)$ .

$f'' (x) = \frac{1458 (- x^{2} + 81)}{{(x^{2} + 243)}^{3}}$

Step 3.17.3.2

Rewrite $81$ as $9^{2}$ .

$f'' (x) = \frac{1458 (- x^{2} + 9^{2})}{{(x^{2} + 243)}^{3}}$

Step 3.17.3.3

Reorder $- x^{2}$ and $9^{2}$ .

$f'' (x) = \frac{1458 (9^{2} - x^{2})}{{(x^{2} + 243)}^{3}}$

Step 3.17.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = x$ .

$f'' (x) = \frac{1458 (9 + x) (9 - x)}{{(x^{2} + 243)}^{3}}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{486 x}{{(x^{2} + 243)}^{2}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

$f' (x) = \frac{(x^{2} + 243) \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} (x^{2} + 243)}{{(x^{2} + 243)}^{2}}$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

$f' (x) = \frac{(x^{2} + 243) (2 x) - x^{2} \frac{d}{d x} (x^{2} + 243)}{{(x^{2} + 243)}^{2}}$

Step 5.1.2.2

Move $2$ to the left of $x^{2} + 243$ .

$f' (x) = \frac{2 \cdot ((x^{2} + 243) x) - x^{2} \frac{d}{d x} (x^{2} + 243)}{{(x^{2} + 243)}^{2}}$

Step 5.1.2.3

By the Sum Rule, the derivative of $x^{2} + 243$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [243]$ .

$f' (x) = \frac{2 (x^{2} + 243) x - x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (243))}{{(x^{2} + 243)}^{2}}$

Step 5.1.2.4

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

$f' (x) = \frac{2 (x^{2} + 243) x - x^{2} (2 x + \frac{d}{d x} (243))}{{(x^{2} + 243)}^{2}}$

Step 5.1.2.5

Since $243$ is constant with respect to $x$ , the derivative of $243$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 (x^{2} + 243) x - x^{2} (2 x + 0)}{{(x^{2} + 243)}^{2}}$

Step 5.1.2.6

Simplify the expression.

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Step 5.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{2 (x^{2} + 243) x - x^{2} (2 x)}{{(x^{2} + 243)}^{2}}$

Step 5.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{2 (x^{2} + 243) x - 2 x^{2} x}{{(x^{2} + 243)}^{2}}$

Step 5.1.3

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

$f' (x) = \frac{2 (x^{2} + 243) x - 2 (x \cdot x^{2})}{{(x^{2} + 243)}^{2}}$

Step 5.1.4

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

$f' (x) = \frac{2 (x^{2} + 243) x - 2 x^{1 + 2}}{{(x^{2} + 243)}^{2}}$

Step 5.1.5

Add $1$ and $2$ .

$f' (x) = \frac{2 (x^{2} + 243) x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6

Simplify.

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Step 5.1.6.1

Apply the distributive property.

$f' (x) = \frac{(2 x^{2} + 2 \cdot 243) x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.2

Apply the distributive property.

$f' (x) = \frac{2 x^{2} x + 2 \cdot (243 x) - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3

Simplify the numerator.

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Step 5.1.6.3.1

Simplify each term.

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Step 5.1.6.3.1.1

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

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Step 5.1.6.3.1.1.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (243 x) - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3.1.1.2

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

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Step 5.1.6.3.1.1.2.1

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

$f' (x) = \frac{2 (x \cdot x^{2}) + 2 \cdot (243 x) - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3.1.1.2.2

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

$f' (x) = \frac{2 x^{1 + 2} + 2 \cdot (243 x) - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3.1.1.3

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} + 2 \cdot (243 x) - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3.1.2

Multiply $2$ by $243$ .

$f' (x) = \frac{2 x^{3} + 486 x - 2 x^{3}}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3.2

Combine the opposite terms in $2 x^{3} + 486 x - 2 x^{3}$ .

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Step 5.1.6.3.2.1

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

$f' (x) = \frac{486 x + 0}{{(x^{2} + 243)}^{2}}$

Step 5.1.6.3.2.2

Add $486 x$ and $0$ .

$f' (x) = \frac{486 x}{{(x^{2} + 243)}^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{486 x}{{(x^{2} + 243)}^{2}}$ .

$\frac{486 x}{{(x^{2} + 243)}^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{486 x}{{(x^{2} + 243)}^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{486 x}{{(x^{2} + 243)}^{2}} = 0$

Step 6.2

Set the numerator equal to zero.

$486 x = 0$

Step 6.3

Divide each term in $486 x = 0$ by $486$ and simplify.

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Step 6.3.1

Divide each term in $486 x = 0$ by $486$ .

$\frac{486 x}{486} = \frac{0}{486}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $486$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{486 x}{486} = \frac{0}{486}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{486}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $0$ by $486$ .

$x = 0$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 8

Critical points to evaluate.

$x = 0$

Step 9

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{1458 (9 + 0) (9 - (0))}{{({(0)}^{2} + 243)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Remove parentheses.

$\frac{1458 (9 + 0) (9 - (0))}{{({(0)}^{2} + 243)}^{3}}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Combine exponents.

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Step 10.2.1.1

Multiply $- 1$ by $0$ .

$\frac{1458 (9 + 0) (9 + 0)}{{(0^{2} + 243)}^{3}}$

Step 10.2.1.2

Raise $9 + 0$ to the power of $1$ .

$\frac{1458 ({(9 + 0)}^{1} (9 + 0))}{{(0^{2} + 243)}^{3}}$

Step 10.2.1.3

Raise $9 + 0$ to the power of $1$ .

$\frac{1458 ({(9 + 0)}^{1} {(9 + 0)}^{1})}{{(0^{2} + 243)}^{3}}$

Step 10.2.1.4

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

$\frac{1458 {(9 + 0)}^{1 + 1}}{{(0^{2} + 243)}^{3}}$

Step 10.2.1.5

Add $1$ and $1$ .

$\frac{1458 {(9 + 0)}^{2}}{{(0^{2} + 243)}^{3}}$

Step 10.2.2

Add $9$ and $0$ .

$\frac{1458 \cdot 9^{2}}{{(0^{2} + 243)}^{3}}$

Step 10.2.3

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

$\frac{1458 \cdot 81}{{(0^{2} + 243)}^{3}}$

Step 10.3

Simplify the denominator.

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Step 10.3.1

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

$\frac{1458 \cdot 81}{{(0 + 243)}^{3}}$

Step 10.3.2

Add $0$ and $243$ .

$\frac{1458 \cdot 81}{243^{3}}$

Step 10.3.3

Raise $243$ to the power of $3$ .

$\frac{1458 \cdot 81}{14348907}$

Step 10.4

Reduce the expression by cancelling the common factors.

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Step 10.4.1

Multiply $1458$ by $81$ .

$\frac{118098}{14348907}$

Step 10.4.2

Cancel the common factor of $118098$ and $14348907$ .

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Step 10.4.2.1

Factor $59049$ out of $118098$ .

$\frac{59049 (2)}{14348907}$

Step 10.4.2.2

Cancel the common factors.

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Step 10.4.2.2.1

Factor $59049$ out of $14348907$ .

$\frac{59049 \cdot 2}{59049 \cdot 243}$

Step 10.4.2.2.2

Cancel the common factor.

$\frac{59049 \cdot 2}{59049 \cdot 243}$

Step 10.4.2.2.3

Rewrite the expression.

$\frac{2}{243}$

Step 11

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 12

Find the y-value when $x = 0$ .

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Step 12.1

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

$f (0) = \frac{{(0)}^{2}}{{(0)}^{2} + 243}$

Step 12.2

Simplify the result.

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Step 12.2.1

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

$f (0) = \frac{0}{0^{2} + 243}$

Step 12.2.2

Simplify the denominator.

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Step 12.2.2.1

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

$f (0) = \frac{0}{0 + 243}$

Step 12.2.2.2

Add $0$ and $243$ .

$f (0) = \frac{0}{243}$

Step 12.2.3

Divide $0$ by $243$ .

$f (0) = 0$

Step 12.2.4

The final answer is $0$ .

$y = 0$

Step 13

These are the local extrema for $f (x) = \frac{x^{2}}{x^{2} + 243}$ .

$(0, 0)$ is a local minima

Step 14