Calculus Examples

$f (x) = \ln (\frac{2 x}{x + 3})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{2 x}{x + 3}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{2 x}{x + 3}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{2 x}{x + 3}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [\frac{2 x}{x + 3}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{2 x}{x + 3}$ .

$\frac{1}{\frac{2 x}{x + 3}} \frac{d}{d x} [\frac{2 x}{x + 3}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{2 x}{x + 3}$ .

$1 \frac{x + 3}{2 x} \frac{d}{d x} [\frac{2 x}{x + 3}]$

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Multiply $\frac{x + 3}{2 x}$ by $1$ .

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

Step 3.2

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

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

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $2$ and $\frac{x + 3}{2 x}$ .

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

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

$\frac{x + 3}{x} \frac{d}{d x} [\frac{x}{x + 3}]$

Step 4

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 $g (x) = x + 3$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

Multiply $x + 3$ by $1$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{x + 3}{x} \cdot \frac{x + 3 - x (1 + 0)}{{(x + 3)}^{2}}$

Step 5.6

Simplify terms.

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Step 5.6.1

Add $1$ and $0$ .

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

Step 5.6.2

Multiply $- 1$ by $1$ .

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

Step 5.6.3

Subtract $x$ from $x$ .

$\frac{x + 3}{x} \cdot \frac{0 + 3}{{(x + 3)}^{2}}$

Step 5.6.4

Add $0$ and $3$ .

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

Step 5.6.5

Multiply $\frac{x + 3}{x}$ by $\frac{3}{{(x + 3)}^{2}}$ .

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

Step 5.6.6

Move $3$ to the left of $x + 3$ .

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

Step 5.6.7

Cancel the common factor of $x + 3$ and ${(x + 3)}^{2}$ .

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Step 5.6.7.1

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

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

Step 5.6.7.2

Cancel the common factors.

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Step 5.6.7.2.1

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

$\frac{(x + 3) \cdot 3}{(x + 3) (x (x + 3))}$

Step 5.6.7.2.2

Cancel the common factor.

$\frac{(x + 3) \cdot 3}{(x + 3) (x (x + 3))}$

Step 5.6.7.2.3

Rewrite the expression.

$\frac{3}{x (x + 3)}$