Calculus Examples

$f (x) = - 2 \ln (4 x)$

Step 1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \ln (4 x)$ with respect to $x$ is $- 2 \frac{d}{d x} [\ln (4 x)]$ .

$- 2 \frac{d}{d x} [\ln (4 x)]$

Step 2

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 $g (x) = 4 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 x$ .

$- 2 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [4 x])$

Step 2.2

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

$- 2 (\frac{1}{u} \frac{d}{d x} [4 x])$

Step 2.3

Replace all occurrences of $u$ with $4 x$ .

$- 2 (\frac{1}{4 x} \frac{d}{d x} [4 x])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{4 x}$ and $- 2$ .

$\frac{- 2}{4 x} \frac{d}{d x} [4 x]$

Step 3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.1

Cancel the common factor of $- 2$ and $4$ .

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Step 3.2.1.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{4 x} \frac{d}{d x} [4 x]$

Step 3.2.1.2

Cancel the common factors.

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Step 3.2.1.2.1

Factor $2$ out of $4 x$ .

$\frac{2 (- 1)}{2 (2 x)} \frac{d}{d x} [4 x]$

Step 3.2.1.2.2

Cancel the common factor.

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

Step 3.2.1.2.3

Rewrite the expression.

$\frac{- 1}{2 x} \frac{d}{d x} [4 x]$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{1}{2 x} \frac{d}{d x} [4 x]$

Step 3.3

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

$- \frac{1}{2 x} (4 \frac{d}{d x} [x])$

Step 3.4

Simplify terms.

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Step 3.4.1

Multiply $4$ by $- 1$ .

$- 4 \frac{1}{2 x} \frac{d}{d x} [x]$

Step 3.4.2

Combine $- 4$ and $\frac{1}{2 x}$ .

$\frac{- 4}{2 x} \frac{d}{d x} [x]$

Step 3.4.3

Cancel the common factor of $- 4$ and $2$ .

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Step 3.4.3.1

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2}{2 x} \frac{d}{d x} [x]$

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

Factor $2$ out of $2 x$ .

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

Step 3.4.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 x} \frac{d}{d x} [x]$

Step 3.4.3.2.3

Rewrite the expression.

$\frac{- 2}{x} \frac{d}{d x} [x]$

Step 3.4.4

Move the negative in front of the fraction.

$- \frac{2}{x} \frac{d}{d x} [x]$

Step 3.5

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

$- \frac{2}{x} \cdot 1$

Step 3.6

Multiply $- 1$ by $1$ .

$- \frac{2}{x}$