Calculus Examples

$y = \ln (3 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (3 x))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $g (x) = 3 x$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u$ as $3 x$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $3 x$ .

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

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

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

$\frac{1}{3 x} (3 \frac{d}{d x} [x])$

Step 3.2.2

Simplify terms.

Tap for more steps...

Step 3.2.2.1

Combine $3$ and $\frac{1}{3 x}$ .

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

Step 3.2.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor.

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

Step 3.2.2.2.2

Rewrite the expression.

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

Step 3.2.3

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

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

Step 3.2.4

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

$\frac{1}{x}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{1}{x}$