Calculus Examples

$f (x) = \ln (5 - x)$

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 $g (x) = 5 - x$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $5 - x$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [5 - x]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [5 - x]$

Step 1.3

Replace all occurrences of $u$ with $5 - x$ .

$\frac{1}{5 - x} \frac{d}{d x} [5 - x]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$\frac{1}{5 - x} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{1}{5 - x} (0 + \frac{d}{d x} [- x])$

Step 2.3

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

$\frac{1}{5 - x} \frac{d}{d x} [- x]$

Step 2.4

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

$\frac{1}{5 - x} (- \frac{d}{d x} [x])$

Step 2.5

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

$\frac{1}{5 - x} (- 1 \cdot 1)$

Step 2.6

Combine fractions.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $1$ .

$\frac{1}{5 - x} \cdot - 1$

Step 2.6.2

Combine $\frac{1}{5 - x}$ and $- 1$ .

$\frac{- 1}{5 - x}$

Step 2.6.3

Move the negative in front of the fraction.

$- \frac{1}{5 - x}$