Find dy/dx y = natural log of 2x

$y = \ln (2 x)$

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\ln (2 x))$

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

$y'$

Differentiate the right side of the equation.

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

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

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

$\frac{1}{u} \frac{d}{d x} [2 x]$

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

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

Differentiate.

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

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

Simplify terms.

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

$\frac{1}{x}$

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

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

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

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

Find dy/dx y = natural log of 2x

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