Calculus Examples

$y = \frac{9 x^{2} + 1}{x^{2} + 9}$

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 $f (x) = 9 x^{2} + 1$ and $g (x) = x^{2} + 9$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [1]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $9$ .

$\frac{(x^{2} + 9) (18 x + \frac{d}{d x} [1]) - (9 x^{2} + 1) \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 2.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 9) (18 x + 0) - (9 x^{2} + 1) \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $18 x$ and $0$ .

$\frac{(x^{2} + 9) (18 x) - (9 x^{2} + 1) \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 2.6.2

Move $18$ to the left of $x^{2} + 9$ .

$\frac{18 \cdot (x^{2} + 9) x - (9 x^{2} + 1) \frac{d}{d x} [x^{2} + 9]}{{(x^{2} + 9)}^{2}}$

Step 2.7

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

$\frac{18 (x^{2} + 9) x - (9 x^{2} + 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])}{{(x^{2} + 9)}^{2}}$

Step 2.8

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

$\frac{18 (x^{2} + 9) x - (9 x^{2} + 1) (2 x + \frac{d}{d x} [9])}{{(x^{2} + 9)}^{2}}$

Step 2.9

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

$\frac{18 (x^{2} + 9) x - (9 x^{2} + 1) (2 x + 0)}{{(x^{2} + 9)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $2 x$ and $0$ .

$\frac{18 (x^{2} + 9) x - (9 x^{2} + 1) (2 x)}{{(x^{2} + 9)}^{2}}$

Step 2.10.2

Multiply $2$ by $- 1$ .

$\frac{18 (x^{2} + 9) x - 2 (9 x^{2} + 1) x}{{(x^{2} + 9)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(18 x^{2} + 18 \cdot 9) x - 2 (9 x^{2} + 1) x}{{(x^{2} + 9)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{18 x^{2} x + 18 \cdot 9 x - 2 (9 x^{2} + 1) x}{{(x^{2} + 9)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{18 x^{2} x + 18 \cdot 9 x + (- 2 (9 x^{2}) - 2 \cdot 1) x}{{(x^{2} + 9)}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{18 x^{2} x + 18 \cdot 9 x - 2 (9 x^{2}) x - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

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

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Step 3.5.1.1.1

Move $x$ .

$\frac{18 (x \cdot x^{2}) + 18 \cdot 9 x - 2 (9 x^{2}) x - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.1.2

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

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Step 3.5.1.1.2.1

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

$\frac{18 (x^{1} x^{2}) + 18 \cdot 9 x - 2 (9 x^{2}) x - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.1.2.2

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

$\frac{18 x^{1 + 2} + 18 \cdot 9 x - 2 (9 x^{2}) x - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.1.3

Add $1$ and $2$ .

$\frac{18 x^{3} + 18 \cdot 9 x - 2 (9 x^{2}) x - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.2

Multiply $18$ by $9$ .

$\frac{18 x^{3} + 162 x - 2 (9 x^{2}) x - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.3

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

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Step 3.5.1.3.1

Move $x$ .

$\frac{18 x^{3} + 162 x - 2 (9 (x \cdot x^{2})) - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.3.2

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

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Step 3.5.1.3.2.1

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

$\frac{18 x^{3} + 162 x - 2 (9 (x^{1} x^{2})) - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.3.2.2

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

$\frac{18 x^{3} + 162 x - 2 (9 x^{1 + 2}) - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.3.3

Add $1$ and $2$ .

$\frac{18 x^{3} + 162 x - 2 (9 x^{3}) - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.4

Multiply $9$ by $- 2$ .

$\frac{18 x^{3} + 162 x - 18 x^{3} - 2 \cdot 1 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.1.5

Multiply $- 2$ by $1$ .

$\frac{18 x^{3} + 162 x - 18 x^{3} - 2 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.2

Combine the opposite terms in $18 x^{3} + 162 x - 18 x^{3} - 2 x$ .

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Step 3.5.2.1

Subtract $18 x^{3}$ from $18 x^{3}$ .

$\frac{162 x + 0 - 2 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.2.2

Add $162 x$ and $0$ .

$\frac{162 x - 2 x}{{(x^{2} + 9)}^{2}}$

Step 3.5.3

Subtract $2 x$ from $162 x$ .

$\frac{160 x}{{(x^{2} + 9)}^{2}}$