# Derivative of $\ln\left(2 x\right)$

The calculator will find the derivative of $\ln\left(2 x\right)$, with steps shown.

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Find $\frac{d}{dx} \left(\ln\left(2 x\right)\right)$.

### Solution

The function $\ln\left(2 x\right)$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = \ln\left(u\right)$ and $g{\left(x \right)} = 2 x$.

Apply the chain rule $\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(2 x\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(2 x\right)\right)}$$

The derivative of the natural logarithm is $\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(2 x\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(2 x\right)$$

$$\frac{\frac{d}{dx} \left(2 x\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(2 x\right)}{{\color{red}\left(2 x\right)}}$$

Apply the constant multiple rule $\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$ with $c = 2$ and $f{\left(x \right)} = x$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)}}{2 x} = \frac{{\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}}{2 x}$$

Apply the power rule $\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$ with $n = 1$, in other words, $\frac{d}{dx} \left(x\right) = 1$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{x} = \frac{{\color{red}\left(1\right)}}{x}$$

Thus, $\frac{d}{dx} \left(\ln\left(2 x\right)\right) = \frac{1}{x}$.

$\frac{d}{dx} \left(\ln\left(2 x\right)\right) = \frac{1}{x}$A