Calculus Examples

$\frac{4 x}{x^{2} + 36}$

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 $f (x) = 4 x$ and $g (x) = x^{2} + 36$ .

$\frac{(x^{2} + 36) \frac{d}{d x} [4 x] - 1 \cdot 4 x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{(x^{2} + 36) (4 \frac{d}{d x} [x]) - 1 \cdot 4 x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.2

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

$\frac{(x^{2} + 36) (4 \cdot 1) - 1 \cdot 4 x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.3

Multiply $4$ by $1$ .

$\frac{(x^{2} + 36) \cdot 4 - 1 \cdot 4 x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.4

By the Sum Rule, the derivative of $x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$ .

$\frac{(x^{2} + 36) \cdot 4 - 1 \cdot 4 x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 2.5

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

$\frac{(x^{2} + 36) \cdot 4 - 1 \cdot 4 x (2 x + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 2.6

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 36) \cdot 4 - 1 \cdot 4 x (2 x + 0)}{{(x^{2} + 36)}^{2}}$

Step 2.7

Add $2 x$ and $0$ .

$\frac{(x^{2} + 36) \cdot 4 - 1 \cdot 4 x (2 x)}{{(x^{2} + 36)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x^{2} \cdot 4 + 36 \cdot 4 - 1 \cdot 4 x (2 x)}{{(x^{2} + 36)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Move $4$ to the left of $x^{2}$ .

$\frac{4 \cdot x^{2} + 36 \cdot 4 - 1 \cdot 4 x (2 x)}{{(x^{2} + 36)}^{2}}$

Step 3.2.1.2

Multiply $36$ by $4$ .

$\frac{4 x^{2} + 144 - 1 \cdot 4 x (2 x)}{{(x^{2} + 36)}^{2}}$

Step 3.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{4 x^{2} + 144 - 1 \cdot 4 \cdot 2 x \cdot x}{{(x^{2} + 36)}^{2}}$

Step 3.2.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.4.1

Move $x$ .

$\frac{4 x^{2} + 144 - 1 \cdot 4 \cdot 2 (x \cdot x)}{{(x^{2} + 36)}^{2}}$

Step 3.2.1.4.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 144 - 1 \cdot 4 \cdot 2 x^{2}}{{(x^{2} + 36)}^{2}}$

Step 3.2.1.5

Multiply $- 1 \cdot 4 \cdot 2$ .

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Step 3.2.1.5.1

Multiply $- 1$ by $4$ .

$\frac{4 x^{2} + 144 - 4 \cdot 2 x^{2}}{{(x^{2} + 36)}^{2}}$

Step 3.2.1.5.2

Multiply $- 4$ by $2$ .

$\frac{4 x^{2} + 144 - 8 x^{2}}{{(x^{2} + 36)}^{2}}$

Step 3.2.2

Subtract $8 x^{2}$ from $4 x^{2}$ .

$\frac{- 4 x^{2} + 144}{{(x^{2} + 36)}^{2}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Factor $4$ out of $- 4 x^{2} + 144$ .

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Step 3.3.1.1

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

$\frac{4 (- x^{2}) + 144}{{(x^{2} + 36)}^{2}}$

Step 3.3.1.2

Factor $4$ out of $144$ .

$\frac{4 (- x^{2}) + 4 (36)}{{(x^{2} + 36)}^{2}}$

Step 3.3.1.3

Factor $4$ out of $4 (- x^{2}) + 4 (36)$ .

$\frac{4 (- x^{2} + 36)}{{(x^{2} + 36)}^{2}}$

Step 3.3.2

Rewrite $36$ as $6^{2}$ .

$\frac{4 (- x^{2} + 6^{2})}{{(x^{2} + 36)}^{2}}$

Step 3.3.3

Reorder $- x^{2}$ and $6^{2}$ .

$\frac{4 (6^{2} - x^{2})}{{(x^{2} + 36)}^{2}}$

Step 3.3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$\frac{4 (6 + x) (6 - x)}{{(x^{2} + 36)}^{2}}$