Calculus Examples

$\frac{x^{2} - 9}{25 - x^{2}}$

Step 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 $f (x) = x^{2} - 9$ and $g (x) = 25 - x^{2}$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$\frac{(25 - x^{2}) (2 x + 0) - (x^{2} - 9) \frac{d}{d x} [25 - x^{2}]}{{(25 - x^{2})}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

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

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

Step 2.5

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

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

Step 2.6

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

$\frac{2 (25 - x^{2}) x - (x^{2} - 9) (0 + \frac{d}{d x} [- x^{2}])}{{(25 - x^{2})}^{2}}$

Step 2.7

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 2.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.9

Multiply.

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Step 2.9.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (25 - x^{2}) x + 1 (x^{2} - 9) \frac{d}{d x} [x^{2}]}{{(25 - x^{2})}^{2}}$

Step 2.9.2

Multiply $x^{2} - 9$ by $1$ .

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

Step 2.10

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

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

Step 2.11

Move $2$ to the left of $x^{2} - 9$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Apply the distributive property.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $2$ by $25$ .

$\frac{50 x + 2 (- x^{2}) x + 2 x^{2} x + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.2

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

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Step 3.5.1.2.1

Move $x$ .

$\frac{50 x + 2 (- (x \cdot x^{2})) + 2 x^{2} x + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.2.2

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

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Step 3.5.1.2.2.1

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

$\frac{50 x + 2 (- (x^{1} x^{2})) + 2 x^{2} x + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.2.2.2

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

$\frac{50 x + 2 (- x^{1 + 2}) + 2 x^{2} x + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.2.3

Add $1$ and $2$ .

$\frac{50 x + 2 (- x^{3}) + 2 x^{2} x + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.3

Multiply $- 1$ by $2$ .

$\frac{50 x - 2 x^{3} + 2 x^{2} x + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.4

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

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Step 3.5.1.4.1

Move $x$ .

$\frac{50 x - 2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.4.2

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

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Step 3.5.1.4.2.1

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

$\frac{50 x - 2 x^{3} + 2 (x^{1} x^{2}) + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.4.2.2

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

$\frac{50 x - 2 x^{3} + 2 x^{1 + 2} + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.4.3

Add $1$ and $2$ .

$\frac{50 x - 2 x^{3} + 2 x^{3} + 2 \cdot - 9 x}{{(25 - x^{2})}^{2}}$

Step 3.5.1.5

Multiply $2$ by $- 9$ .

$\frac{50 x - 2 x^{3} + 2 x^{3} - 18 x}{{(25 - x^{2})}^{2}}$

Step 3.5.2

Combine the opposite terms in $50 x - 2 x^{3} + 2 x^{3} - 18 x$ .

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Step 3.5.2.1

Add $- 2 x^{3}$ and $2 x^{3}$ .

$\frac{50 x + 0 - 18 x}{{(25 - x^{2})}^{2}}$

Step 3.5.2.2

Add $50 x$ and $0$ .

$\frac{50 x - 18 x}{{(25 - x^{2})}^{2}}$

Step 3.5.3

Subtract $18 x$ from $50 x$ .

$\frac{32 x}{{(25 - x^{2})}^{2}}$

Step 3.6

Reorder terms.

$\frac{32 x}{{(- x^{2} + 25)}^{2}}$

Step 3.7

Simplify the denominator.

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Step 3.7.1

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

$\frac{32 x}{{(- x^{2} + 5^{2})}^{2}}$

Step 3.7.2

Reorder $- x^{2}$ and $5^{2}$ .

$\frac{32 x}{{(5^{2} - x^{2})}^{2}}$

Step 3.7.3

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

$\frac{32 x}{{((5 + x) (5 - x))}^{2}}$

Step 3.7.4

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

$\frac{32 x}{{(5 + x)}^{2} {(5 - x)}^{2}}$