Calculus Examples

$4 \ln (- 2 - x)$

Step 1

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

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

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 $g (x) = - 2 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- 2 - x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- 2 - x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{4}{- 2 - x} (0 + \frac{d}{d x} [- x])$

Step 3.4

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

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

Step 3.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 3.6

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

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

Step 3.7

Combine fractions.

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Step 3.7.1

Multiply $- 1$ by $1$ .

$\frac{4}{- 2 - x} \cdot - 1$

Step 3.7.2

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

$\frac{4 \cdot - 1}{- 2 - x}$

Step 3.7.3

Simplify the expression.

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Step 3.7.3.1

Multiply $4$ by $- 1$ .

$\frac{- 4}{- 2 - x}$

Step 3.7.3.2

Move the negative in front of the fraction.

$- \frac{4}{- 2 - x}$

Step 4

Simplify.

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Step 4.1

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 4.2

Factor $- 1$ out of $- x$ .

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

Step 4.3

Factor $- 1$ out of $- 1 (2) - (x)$ .

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

Step 4.4

Move the negative in front of the fraction.

$- - \frac{4}{2 + x}$

Step 4.5

Multiply $- 1$ by $- 1$ .

$1 \frac{4}{2 + x}$

Step 4.6

Multiply $\frac{4}{2 + x}$ by $1$ .

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