Calculus Examples

$\frac{4 x}{3 x^{2} + 27}$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 x}{3 x^{2} + 27}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{x}{3 x^{2} + 27}]$ .

$4 \frac{d}{d x} [\frac{x}{3 x^{2} + 27}]$

Step 2

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 $f (x) = x$ and $g (x) = 3 x^{2} + 27$ .

$4 \frac{(3 x^{2} + 27) \frac{d}{d x} [x] - x \frac{d}{d x} [3 x^{2} + 27]}{{(3 x^{2} + 27)}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

$4 \frac{(3 x^{2} + 27) \cdot 1 - x \frac{d}{d x} [3 x^{2} + 27]}{{(3 x^{2} + 27)}^{2}}$

Step 3.2

Multiply $3 x^{2} + 27$ by $1$ .

$4 \frac{3 x^{2} + 27 - x \frac{d}{d x} [3 x^{2} + 27]}{{(3 x^{2} + 27)}^{2}}$

Step 3.3

By the Sum Rule, the derivative of $3 x^{2} + 27$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [27]$ .

$4 \frac{3 x^{2} + 27 - x (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [27])}{{(3 x^{2} + 27)}^{2}}$

Step 3.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$4 \frac{3 x^{2} + 27 - x (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [27])}{{(3 x^{2} + 27)}^{2}}$

Step 3.5

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

$4 \frac{3 x^{2} + 27 - x (3 (2 x) + \frac{d}{d x} [27])}{{(3 x^{2} + 27)}^{2}}$

Step 3.6

Multiply $2$ by $3$ .

$4 \frac{3 x^{2} + 27 - x (6 x + \frac{d}{d x} [27])}{{(3 x^{2} + 27)}^{2}}$

Step 3.7

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$4 \frac{3 x^{2} + 27 - x (6 x + 0)}{{(3 x^{2} + 27)}^{2}}$

Step 3.8

Simplify the expression.

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Step 3.8.1

Add $6 x$ and $0$ .

$4 \frac{3 x^{2} + 27 - x (6 x)}{{(3 x^{2} + 27)}^{2}}$

Step 3.8.2

Multiply $6$ by $- 1$ .

$4 \frac{3 x^{2} + 27 - 6 x \cdot x}{{(3 x^{2} + 27)}^{2}}$

Step 4

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

$4 \frac{3 x^{2} + 27 - 6 (x^{1} x)}{{(3 x^{2} + 27)}^{2}}$

Step 5

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

$4 \frac{3 x^{2} + 27 - 6 (x^{1} x^{1})}{{(3 x^{2} + 27)}^{2}}$

Step 6

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

$4 \frac{3 x^{2} + 27 - 6 x^{1 + 1}}{{(3 x^{2} + 27)}^{2}}$

Step 7

Add $1$ and $1$ .

$4 \frac{3 x^{2} + 27 - 6 x^{2}}{{(3 x^{2} + 27)}^{2}}$

Step 8

Subtract $6 x^{2}$ from $3 x^{2}$ .

$4 \frac{- 3 x^{2} + 27}{{(3 x^{2} + 27)}^{2}}$

Step 9

Combine $4$ and $\frac{- 3 x^{2} + 27}{{(3 x^{2} + 27)}^{2}}$ .

$\frac{4 (- 3 x^{2} + 27)}{{(3 x^{2} + 27)}^{2}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{4 (- 3 x^{2}) + 4 \cdot 27}{{(3 x^{2} + 27)}^{2}}$

Step 10.2

Simplify each term.

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Step 10.2.1

Multiply $- 3$ by $4$ .

$\frac{- 12 x^{2} + 4 \cdot 27}{{(3 x^{2} + 27)}^{2}}$

Step 10.2.2

Multiply $4$ by $27$ .

$\frac{- 12 x^{2} + 108}{{(3 x^{2} + 27)}^{2}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Factor $12$ out of $- 12 x^{2} + 108$ .

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Step 10.3.1.1

Factor $12$ out of $- 12 x^{2}$ .

$\frac{12 (- x^{2}) + 108}{{(3 x^{2} + 27)}^{2}}$

Step 10.3.1.2

Factor $12$ out of $108$ .

$\frac{12 (- x^{2}) + 12 (9)}{{(3 x^{2} + 27)}^{2}}$

Step 10.3.1.3

Factor $12$ out of $12 (- x^{2}) + 12 (9)$ .

$\frac{12 (- x^{2} + 9)}{{(3 x^{2} + 27)}^{2}}$

Step 10.3.2

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

$\frac{12 (- x^{2} + 3^{2})}{{(3 x^{2} + 27)}^{2}}$

Step 10.3.3

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

$\frac{12 (3^{2} - x^{2})}{{(3 x^{2} + 27)}^{2}}$

Step 10.3.4

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

$\frac{12 (3 + x) (3 - x)}{{(3 x^{2} + 27)}^{2}}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Factor $3$ out of $3 x^{2} + 27$ .

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Step 10.4.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{12 (3 + x) (3 - x)}{{(3 (x^{2}) + 27)}^{2}}$

Step 10.4.1.2

Factor $3$ out of $27$ .

$\frac{12 (3 + x) (3 - x)}{{(3 x^{2} + 3 \cdot 9)}^{2}}$

Step 10.4.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot 9$ .

$\frac{12 (3 + x) (3 - x)}{{(3 (x^{2} + 9))}^{2}}$

Step 10.4.2

Apply the product rule to $3 (x^{2} + 9)$ .

$\frac{12 (3 + x) (3 - x)}{3^{2} {(x^{2} + 9)}^{2}}$

Step 10.4.3

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

$\frac{12 (3 + x) (3 - x)}{9 {(x^{2} + 9)}^{2}}$

Step 10.5

Cancel the common factor of $12$ and $9$ .

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Step 10.5.1

Factor $3$ out of $12 (3 + x) (3 - x)$ .

$\frac{3 (4 (3 + x) (3 - x))}{9 {(x^{2} + 9)}^{2}}$

Step 10.5.2

Cancel the common factors.

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Step 10.5.2.1

Factor $3$ out of $9 {(x^{2} + 9)}^{2}$ .

$\frac{3 (4 (3 + x) (3 - x))}{3 (3 {(x^{2} + 9)}^{2})}$

Step 10.5.2.2

Cancel the common factor.

$\frac{3 (4 (3 + x) (3 - x))}{3 (3 {(x^{2} + 9)}^{2})}$

Step 10.5.2.3

Rewrite the expression.

$\frac{4 (3 + x) (3 - x)}{3 {(x^{2} + 9)}^{2}}$