Calculus Examples

$\frac{4 x b}{2 a - 6 x}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Cancel the common factor of $4$ and $2 a - 6 x$ .

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Step 1.1.1

Factor $2$ out of $4 x b$ .

$\frac{d}{d x} [\frac{2 (2 x b)}{2 a - 6 x}]$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $2 a$ .

$\frac{d}{d x} [\frac{2 (2 x b)}{2 (a) - 6 x}]$

Step 1.1.2.2

Factor $2$ out of $- 6 x$ .

$\frac{d}{d x} [\frac{2 (2 x b)}{2 (a) + 2 (- 3 x)}]$

Step 1.1.2.3

Factor $2$ out of $2 (a) + 2 (- 3 x)$ .

$\frac{d}{d x} [\frac{2 (2 x b)}{2 (a - 3 x)}]$

Step 1.1.2.4

Cancel the common factor.

$\frac{d}{d x} [\frac{2 (2 x b)}{2 (a - 3 x)}]$

Step 1.1.2.5

Rewrite the expression.

$\frac{d}{d x} [\frac{2 x b}{a - 3 x}]$

Step 1.2

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

$2 b \frac{d}{d x} [\frac{x}{a - 3 x}]$

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 $g (x) = a - 3 x$ .

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

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 $n = 1$ .

$2 b \frac{(a - 3 x) \cdot 1 - x \frac{d}{d x} [a - 3 x]}{{(a - 3 x)}^{2}}$

Step 3.2

Multiply $a - 3 x$ by $1$ .

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

Step 3.3

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

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

Step 3.4

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

$2 b \frac{a - 3 x - x (0 + \frac{d}{d x} [- 3 x])}{{(a - 3 x)}^{2}}$

Step 3.5

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

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

Step 3.6

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

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

Step 3.7

Multiply $- 3$ by $- 1$ .

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

Step 3.8

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

$2 b \frac{a - 3 x + 3 x \cdot 1}{{(a - 3 x)}^{2}}$

Step 3.9

Simplify terms.

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Step 3.9.1

Multiply $3$ by $1$ .

$2 b \frac{a - 3 x + 3 x}{{(a - 3 x)}^{2}}$

Step 3.9.2

Add $- 3 x$ and $3 x$ .

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

Step 3.9.3

Add $a$ and $0$ .

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

Step 3.9.4

Combine $2$ and $\frac{a}{{(a - 3 x)}^{2}}$ .

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

Step 3.9.5

Combine $b$ and $\frac{2 a}{{(a - 3 x)}^{2}}$ .

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

Step 3.9.6

Reorder.

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Step 3.9.6.1

Move $2$ to the left of $b$ .

$\frac{2 \cdot b a}{{(a - 3 x)}^{2}}$

Step 3.9.6.2

Reorder terms.

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