Calculus Examples

$f (x) = {(- 9 x^{2} + 25)}^{2}$

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 chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = - 9 x^{2} + 25$ .

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Step 1.1.1.1

To apply the Chain Rule, set $u$ as $- 9 x^{2} + 25$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [- 9 x^{2} + 25]$

Step 1.1.1.2

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

$2 u \frac{d}{d x} [- 9 x^{2} + 25]$

Step 1.1.1.3

Replace all occurrences of $u$ with $- 9 x^{2} + 25$ .

$2 (- 9 x^{2} + 25) \frac{d}{d x} [- 9 x^{2} + 25]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- 9 x^{2} + 25$ with respect to $x$ is $\frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [25]$ .

$2 (- 9 x^{2} + 25) (\frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [25])$

Step 1.1.2.2

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$2 (- 9 x^{2} + 25) (- 9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [25])$

Step 1.1.2.3

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

$2 (- 9 x^{2} + 25) (- 9 (2 x) + \frac{d}{d x} [25])$

Step 1.1.2.4

Multiply $2$ by $- 9$ .

$2 (- 9 x^{2} + 25) (- 18 x + \frac{d}{d x} [25])$

Step 1.1.2.5

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$2 (- 9 x^{2} + 25) (- 18 x + 0)$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $- 18 x$ and $0$ .

$2 (- 9 x^{2} + 25) (- 18 x)$

Step 1.1.2.6.2

Multiply $- 18$ by $2$ .

$- 36 (- 9 x^{2} + 25) x$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$(- 36 (- 9 x^{2}) - 36 \cdot 25) x$

Step 1.1.3.2

Apply the distributive property.

$- 36 (- 9 x^{2}) x - 36 \cdot 25 x$

Step 1.1.3.3

Combine terms.

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Step 1.1.3.3.1

Multiply $- 9$ by $- 36$ .

$324 x^{2} x - 36 \cdot 25 x$

Step 1.1.3.3.2

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

$324 (x^{1} x^{2}) - 36 \cdot 25 x$

Step 1.1.3.3.3

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

$324 x^{1 + 2} - 36 \cdot 25 x$

Step 1.1.3.3.4

Add $1$ and $2$ .

$324 x^{3} - 36 \cdot 25 x$

Step 1.1.3.3.5

Multiply $- 36$ by $25$ .

$f' (x) = 324 x^{3} - 900 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $324 x^{3} - 900 x$ .

$324 x^{3} - 900 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $324 x^{3} - 900 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$324 x^{3} - 900 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $36 x$ out of $324 x^{3} - 900 x$ .

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Step 2.2.1.1

Factor $36 x$ out of $324 x^{3}$ .

$36 x (9 x^{2}) - 900 x = 0$

Step 2.2.1.2

Factor $36 x$ out of $- 900 x$ .

$36 x (9 x^{2}) + 36 x (- 25) = 0$

Step 2.2.1.3

Factor $36 x$ out of $36 x (9 x^{2}) + 36 x (- 25)$ .

$36 x (9 x^{2} - 25) = 0$

Step 2.2.2

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$36 x ({(3 x)}^{2} - 25) = 0$

Step 2.2.3

Rewrite $25$ as $5^{2}$ .

$36 x ({(3 x)}^{2} - 5^{2}) = 0$

Step 2.2.4

Factor.

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Step 2.2.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 5$ .

$36 x ((3 x + 5) (3 x - 5)) = 0$

Step 2.2.4.2

Remove unnecessary parentheses.

$36 x (3 x + 5) (3 x - 5) = 0$

Step 2.3

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

$x = 0$

$3 x + 5 = 0$

$3 x - 5 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 2.5.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 2.5.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 2.6

Set $3 x - 5$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Step 2.6.2

Solve $3 x - 5 = 0$ for $x$ .

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Step 2.6.2.1

Add $5$ to both sides of the equation.

$3 x = 5$

Step 2.6.2.2

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 2.6.2.2.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 2.6.2.2.2

Simplify the left side.

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Step 2.6.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.6.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 2.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 2.7

The final solution is all the values that make $36 x (3 x + 5) (3 x - 5) = 0$ true.

$x = 0, - \frac{5}{3}, \frac{5}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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 4

Evaluate ${(- 9 x^{2} + 25)}^{2}$ 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 $x$ .

${(- 9 {(0)}^{2} + 25)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

${(- 9 \cdot 0 + 25)}^{2}$

Step 4.1.2.1.2

Multiply $- 9$ by $0$ .

${(0 + 25)}^{2}$

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Add $0$ and $25$ .

$25^{2}$

Step 4.1.2.2.2

Raise $25$ to the power of $2$ .

$625$

Step 4.2

Evaluate at $x = - \frac{5}{3}$ .

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Step 4.2.1

Substitute $- \frac{5}{3}$ for $x$ .

${(- 9 {(- \frac{5}{3})}^{2} + 25)}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.1.1

Apply the product rule to $- \frac{5}{3}$ .

${(- 9 ({(- 1)}^{2} {(\frac{5}{3})}^{2}) + 25)}^{2}$

Step 4.2.2.1.1.2

Apply the product rule to $\frac{5}{3}$ .

${(- 9 ({(- 1)}^{2} \frac{5^{2}}{3^{2}}) + 25)}^{2}$

Step 4.2.2.1.2

Raise $- 1$ to the power of $2$ .

${(- 9 (1 \frac{5^{2}}{3^{2}}) + 25)}^{2}$

Step 4.2.2.1.3

Multiply $\frac{5^{2}}{3^{2}}$ by $1$ .

${(- 9 \frac{5^{2}}{3^{2}} + 25)}^{2}$

Step 4.2.2.1.4

Raise $5$ to the power of $2$ .

${(- 9 \frac{25}{3^{2}} + 25)}^{2}$

Step 4.2.2.1.5

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

${(- 9 (\frac{25}{9}) + 25)}^{2}$

Step 4.2.2.1.6

Cancel the common factor of $9$ .

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Step 4.2.2.1.6.1

Factor $9$ out of $- 9$ .

${(9 (- 1) \frac{25}{9} + 25)}^{2}$

Step 4.2.2.1.6.2

Cancel the common factor.

${(9 \cdot - 1 \frac{25}{9} + 25)}^{2}$

Step 4.2.2.1.6.3

Rewrite the expression.

${(- 1 \cdot 25 + 25)}^{2}$

Step 4.2.2.1.7

Multiply $- 1$ by $25$ .

${(- 25 + 25)}^{2}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Add $- 25$ and $25$ .

$0^{2}$

Step 4.2.2.2.2

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

$0$

Step 4.3

Evaluate at $x = \frac{5}{3}$ .

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Step 4.3.1

Substitute $\frac{5}{3}$ for $x$ .

${(- 9 {(\frac{5}{3})}^{2} + 25)}^{2}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Apply the product rule to $\frac{5}{3}$ .

${(- 9 \frac{5^{2}}{3^{2}} + 25)}^{2}$

Step 4.3.2.1.2

Raise $5$ to the power of $2$ .

${(- 9 \frac{25}{3^{2}} + 25)}^{2}$

Step 4.3.2.1.3

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

${(- 9 (\frac{25}{9}) + 25)}^{2}$

Step 4.3.2.1.4

Cancel the common factor of $9$ .

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Step 4.3.2.1.4.1

Factor $9$ out of $- 9$ .

${(9 (- 1) \frac{25}{9} + 25)}^{2}$

Step 4.3.2.1.4.2

Cancel the common factor.

${(9 \cdot - 1 \frac{25}{9} + 25)}^{2}$

Step 4.3.2.1.4.3

Rewrite the expression.

${(- 1 \cdot 25 + 25)}^{2}$

Step 4.3.2.1.5

Multiply $- 1$ by $25$ .

${(- 25 + 25)}^{2}$

Step 4.3.2.2

Simplify the expression.

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Step 4.3.2.2.1

Add $- 25$ and $25$ .

$0^{2}$

Step 4.3.2.2.2

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

$0$

Step 4.4

List all of the points.

$(0, 625), (- \frac{5}{3}, 0), (\frac{5}{3}, 0)$

Step 5