Calculus Examples

$f (x) = \frac{x^{3}}{(x^{2}) - 25}$

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 $f (x) = x^{3}$ and $g (x) = x^{2} - 25$ .

$\frac{(x^{2} - 25) \frac{d}{d x} [x^{3}] - x^{3} \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

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 $n = 3$ .

$\frac{(x^{2} - 25) (3 x^{2}) - x^{3} \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 1.1.2.2

Move $3$ to the left of $x^{2} - 25$ .

$\frac{3 \cdot (x^{2} - 25) x^{2} - x^{3} \frac{d}{d x} [x^{2} - 25]}{{(x^{2} - 25)}^{2}}$

Step 1.1.2.3

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

$\frac{3 (x^{2} - 25) x^{2} - x^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25])}{{(x^{2} - 25)}^{2}}$

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 $n = 2$ .

$\frac{3 (x^{2} - 25) x^{2} - x^{3} (2 x + \frac{d}{d x} [- 25])}{{(x^{2} - 25)}^{2}}$

Step 1.1.2.5

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

$\frac{3 (x^{2} - 25) x^{2} - x^{3} (2 x + 0)}{{(x^{2} - 25)}^{2}}$

Step 1.1.2.6

Simplify the expression.

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Step 1.1.2.6.1

Add $2 x$ and $0$ .

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

Step 1.1.2.6.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3

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

$\frac{3 (x^{2} - 25) x^{2} - 2 (x^{1} x^{3})}{{(x^{2} - 25)}^{2}}$

Step 1.1.4

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

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

Step 1.1.5

Add $1$ and $3$ .

$\frac{3 (x^{2} - 25) x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Apply the distributive property.

$\frac{(3 x^{2} + 3 \cdot - 25) x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.2

Apply the distributive property.

$\frac{3 x^{2} x^{2} + 3 \cdot - 25 x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

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

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Step 1.1.6.3.1.1.1

Move $x^{2}$ .

$\frac{3 (x^{2} x^{2}) + 3 \cdot - 25 x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.3.1.1.2

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

$\frac{3 x^{2 + 2} + 3 \cdot - 25 x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.3.1.1.3

Add $2$ and $2$ .

$\frac{3 x^{4} + 3 \cdot - 25 x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.3.1.2

Multiply $3$ by $- 25$ .

$\frac{3 x^{4} - 75 x^{2} - 2 x^{4}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$\frac{x^{4} - 75 x^{2}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.4

Factor $x^{2}$ out of $x^{4} - 75 x^{2}$ .

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Step 1.1.6.4.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{x^{2} x^{2} - 75 x^{2}}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.4.2

Factor $x^{2}$ out of $- 75 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} \cdot - 75}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 75$ .

$\frac{x^{2} (x^{2} - 75)}{{(x^{2} - 25)}^{2}}$

Step 1.1.6.5

Simplify the denominator.

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Step 1.1.6.5.1

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

$\frac{x^{2} (x^{2} - 75)}{{(x^{2} - 5^{2})}^{2}}$

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 $b = 5$ .

$\frac{x^{2} (x^{2} - 75)}{{((x + 5) (x - 5))}^{2}}$

Step 1.1.6.5.3

Apply the product rule to $(x + 5) (x - 5)$ .

$f' (x) = \frac{x^{2} (x^{2} - 75)}{{(x + 5)}^{2} {(x - 5)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} (x^{2} - 75)}{{(x + 5)}^{2} {(x - 5)}^{2}}$ .

$\frac{x^{2} (x^{2} - 75)}{{(x + 5)}^{2} {(x - 5)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} (x^{2} - 75)}{{(x + 5)}^{2} {(x - 5)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{x^{2} (x^{2} - 75)}{{(x + 5)}^{2} {(x - 5)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$x^{2} (x^{2} - 75) = 0$

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 $0$ .

$x^{2} = 0$

$x^{2} - 75 = 0$

Step 2.3.2

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.3.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.3.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.3.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 2.3.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.3.3

Set $x^{2} - 75$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

Set $x^{2} - 75$ equal to $0$ .

$x^{2} - 75 = 0$

Step 2.3.3.2

Solve $x^{2} - 75 = 0$ for $x$ .

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Step 2.3.3.2.1

Add $75$ to both sides of the equation.

$x^{2} = 75$

Step 2.3.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{75}$

Step 2.3.3.2.3

Simplify $\pm \sqrt{75}$ .

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Step 2.3.3.2.3.1

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 2.3.3.2.3.1.1

Factor $25$ out of $75$ .

$x = \pm \sqrt{25 (3)}$

Step 2.3.3.2.3.1.2

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

$x = \pm \sqrt{5^{2} \cdot 3}$

Step 2.3.3.2.3.2

Pull terms out from under the radical.

$x = \pm 5 \sqrt{3}$

Step 2.3.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5 \sqrt{3}$

Step 2.3.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5 \sqrt{3}$

Step 2.3.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5 \sqrt{3}, - 5 \sqrt{3}$

Step 2.3.4

The final solution is all the values that make $x^{2} (x^{2} - 75) = 0$ true.

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

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{x^{2} (x^{2} - 75)}{{(x + 5)}^{2} {(x - 5)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

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 $0$ .

${(x + 5)}^{2} = 0$

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

Step 3.2.2

Set ${(x + 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.2.1

Set ${(x + 5)}^{2}$ equal to $0$ .

${(x + 5)}^{2} = 0$

Step 3.2.2.2

Solve ${(x + 5)}^{2} = 0$ for $x$ .

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Step 3.2.2.2.1

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

$x + 5 = 0$

Step 3.2.2.2.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3.2.3

Set ${(x - 5)}^{2}$ equal to $0$ and solve for $x$ .

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Step 3.2.3.1

Set ${(x - 5)}^{2}$ equal to $0$ .

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

Step 3.2.3.2

Solve ${(x - 5)}^{2} = 0$ for $x$ .

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Step 3.2.3.2.1

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

$x - 5 = 0$

Step 3.2.3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 3.2.4

The final solution is all the values that make ${(x + 5)}^{2} {(x - 5)}^{2} = 0$ true.

$x = - 5, 5$

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 $0$ .

$x = - 5, x = 5$

Step 4

Evaluate $\frac{x^{3}}{(x^{2}) - 25}$ 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 $x$ .

$\frac{{(0)}^{3}}{({(0)}^{2}) - 25}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

$\frac{0}{0^{2} - 25}$

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 $0$ .

$\frac{0}{0 - 25}$

Step 4.1.2.2.2

Subtract $25$ from $0$ .

$\frac{0}{- 25}$

Step 4.1.2.3

Divide $0$ by $- 25$ .

$0$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

Raise $5$ to the power of $3$ .

$\frac{125 {\sqrt{3}}^{3}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.1.3

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{125 \sqrt{3^{3}}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.1.4

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

$\frac{125 \sqrt{27}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.1.5

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.2.1.5.1

Factor $9$ out of $27$ .

$\frac{125 \sqrt{9 (3)}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.1.5.2

Rewrite $9$ as $3^{2}$ .

$\frac{125 \sqrt{3^{2} \cdot 3}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.1.6

Pull terms out from under the radical.

$\frac{125 \cdot 3 \sqrt{3}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.1.7

Multiply $125$ by $3$ .

$\frac{375 \sqrt{3}}{{(5 \sqrt{3})}^{2} - 25}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

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

$\frac{375 \sqrt{3}}{5^{2} {\sqrt{3}}^{2} - 25}$

Step 4.2.2.2.2

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

$\frac{375 \sqrt{3}}{25 {\sqrt{3}}^{2} - 25}$

Step 4.2.2.2.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{375 \sqrt{3}}{25 {(3^{\frac{1}{2}})}^{2} - 25}$

Step 4.2.2.2.3.2

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

$\frac{375 \sqrt{3}}{25 \cdot 3^{\frac{1}{2} \cdot 2} - 25}$

Step 4.2.2.2.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{375 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} - 25}$

Step 4.2.2.2.3.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.3.4.1

Cancel the common factor.

$\frac{375 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} - 25}$

Step 4.2.2.2.3.4.2

Rewrite the expression.

$\frac{375 \sqrt{3}}{25 \cdot 3^{1} - 25}$

Step 4.2.2.2.3.5

Evaluate the exponent.

$\frac{375 \sqrt{3}}{25 \cdot 3 - 25}$

Step 4.2.2.2.4

Multiply $25$ by $3$ .

$\frac{375 \sqrt{3}}{75 - 25}$

Step 4.2.2.2.5

Subtract $25$ from $75$ .

$\frac{375 \sqrt{3}}{50}$

Step 4.2.2.3

Cancel the common factor of $375$ and $50$ .

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Step 4.2.2.3.1

Factor $25$ out of $375 \sqrt{3}$ .

$\frac{25 (15 \sqrt{3})}{50}$

Step 4.2.2.3.2

Cancel the common factors.

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Step 4.2.2.3.2.1

Factor $25$ out of $50$ .

$\frac{25 (15 \sqrt{3})}{25 (2)}$

Step 4.2.2.3.2.2

Cancel the common factor.

$\frac{25 (15 \sqrt{3})}{25 \cdot 2}$

Step 4.2.2.3.2.3

Rewrite the expression.

$\frac{15 \sqrt{3}}{2}$

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the numerator.

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Step 4.3.2.1.1

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

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

Step 4.3.2.1.2

Raise $- 5$ to the power of $3$ .

$\frac{- 125 {\sqrt{3}}^{3}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.1.3

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{- 125 \sqrt{3^{3}}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.1.4

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

$\frac{- 125 \sqrt{27}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.1.5

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.3.2.1.5.1

Factor $9$ out of $27$ .

$\frac{- 125 \sqrt{9 (3)}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.1.5.2

Rewrite $9$ as $3^{2}$ .

$\frac{- 125 \sqrt{3^{2} \cdot 3}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.1.6

Pull terms out from under the radical.

$\frac{- 125 \cdot 3 \sqrt{3}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.1.7

Multiply $- 125$ by $3$ .

$\frac{- 375 \sqrt{3}}{{(- 5 \sqrt{3})}^{2} - 25}$

Step 4.3.2.2

Simplify the denominator.

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Step 4.3.2.2.1

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

$\frac{- 375 \sqrt{3}}{{(- 5)}^{2} {\sqrt{3}}^{2} - 25}$

Step 4.3.2.2.2

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

$\frac{- 375 \sqrt{3}}{25 {\sqrt{3}}^{2} - 25}$

Step 4.3.2.2.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{- 375 \sqrt{3}}{25 {(3^{\frac{1}{2}})}^{2} - 25}$

Step 4.3.2.2.3.2

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

$\frac{- 375 \sqrt{3}}{25 \cdot 3^{\frac{1}{2} \cdot 2} - 25}$

Step 4.3.2.2.3.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{- 375 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} - 25}$

Step 4.3.2.2.3.4

Cancel the common factor of $2$ .

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Step 4.3.2.2.3.4.1

Cancel the common factor.

$\frac{- 375 \sqrt{3}}{25 \cdot 3^{\frac{2}{2}} - 25}$

Step 4.3.2.2.3.4.2

Rewrite the expression.

$\frac{- 375 \sqrt{3}}{25 \cdot 3^{1} - 25}$

Step 4.3.2.2.3.5

Evaluate the exponent.

$\frac{- 375 \sqrt{3}}{25 \cdot 3 - 25}$

Step 4.3.2.2.4

Multiply $25$ by $3$ .

$\frac{- 375 \sqrt{3}}{75 - 25}$

Step 4.3.2.2.5

Subtract $25$ from $75$ .

$\frac{- 375 \sqrt{3}}{50}$

Step 4.3.2.3

Reduce the expression by cancelling the common factors.

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Step 4.3.2.3.1

Cancel the common factor of $- 375$ and $50$ .

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Step 4.3.2.3.1.1

Factor $25$ out of $- 375 \sqrt{3}$ .

$\frac{25 (- 15 \sqrt{3})}{50}$

Step 4.3.2.3.1.2

Cancel the common factors.

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Step 4.3.2.3.1.2.1

Factor $25$ out of $50$ .

$\frac{25 (- 15 \sqrt{3})}{25 (2)}$

Step 4.3.2.3.1.2.2

Cancel the common factor.

$\frac{25 (- 15 \sqrt{3})}{25 \cdot 2}$

Step 4.3.2.3.1.2.3

Rewrite the expression.

$\frac{- 15 \sqrt{3}}{2}$

Step 4.3.2.3.2

Move the negative in front of the fraction.

$- \frac{15 \sqrt{3}}{2}$

Step 4.4

Evaluate at $x = - 5$ .

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Step 4.4.1

Substitute $- 5$ for $x$ .

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

Step 4.4.2

Simplify.

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Step 4.4.2.1

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

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

Step 4.4.2.2

Subtract $25$ from $25$ .

$\frac{{(- 5)}^{3}}{0}$

Step 4.4.2.3

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

Undefined

Step 4.5

Evaluate at $x = 5$ .

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Step 4.5.1

Substitute $5$ for $x$ .

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

Step 4.5.2

Simplify.

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Step 4.5.2.1

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

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

Step 4.5.2.2

Subtract $25$ from $25$ .

$\frac{{(5)}^{3}}{0}$

Step 4.5.2.3

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

Undefined

Step 4.6

List all of the points.

$(0, 0), (5 \sqrt{3}, \frac{15 \sqrt{3}}{2}), (- 5 \sqrt{3}, - \frac{15 \sqrt{3}}{2})$

Step 5