Calculus Examples

$25 \ln (x + 2) + 14$

Step 1

By the Sum Rule, the derivative of $25 \ln (x + 2) + 14$ with respect to $x$ is $\frac{d}{d x} [25 \ln (x + 2)] + \frac{d}{d x} [14]$ .

$\frac{d}{d x} [25 \ln (x + 2)] + \frac{d}{d x} [14]$

Step 2

Evaluate $\frac{d}{d x} [25 \ln (x + 2)]$ .

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Step 2.1

Since $25$ is constant with respect to $x$ , the derivative of $25 \ln (x + 2)$ with respect to $x$ is $25 \frac{d}{d x} [\ln (x + 2)]$ .

$25 \frac{d}{d x} [\ln (x + 2)] + \frac{d}{d x} [14]$

Step 2.2

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 $g (x) = x + 2$ .

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Step 2.2.1

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

$25 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x + 2]) + \frac{d}{d x} [14]$

Step 2.2.2

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

$25 (\frac{1}{u} \frac{d}{d x} [x + 2]) + \frac{d}{d x} [14]$

Step 2.2.3

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

$25 (\frac{1}{x + 2} \frac{d}{d x} [x + 2]) + \frac{d}{d x} [14]$

Step 2.3

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

$25 (\frac{1}{x + 2} (\frac{d}{d x} [x] + \frac{d}{d x} [2])) + \frac{d}{d x} [14]$

Step 2.4

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

$25 (\frac{1}{x + 2} (1 + \frac{d}{d x} [2])) + \frac{d}{d x} [14]$

Step 2.5

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

$25 (\frac{1}{x + 2} (1 + 0)) + \frac{d}{d x} [14]$

Step 2.6

Add $1$ and $0$ .

$25 (\frac{1}{x + 2} \cdot 1) + \frac{d}{d x} [14]$

Step 2.7

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

$25 \frac{1}{x + 2} + \frac{d}{d x} [14]$

Step 2.8

Combine $25$ and $\frac{1}{x + 2}$ .

$\frac{25}{x + 2} + \frac{d}{d x} [14]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $14$ is constant with respect to $x$ , the derivative of $14$ with respect to $x$ is $0$ .

$\frac{25}{x + 2} + 0$

Step 3.2

Add $\frac{25}{x + 2}$ and $0$ .

$\frac{25}{x + 2}$