# Second derivative of $\ln\left(5 x\right)$

The calculator will find the second derivative of $\ln\left(5 x\right)$, with steps shown.

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Find $\frac{d^{2}}{dx^{2}} \left(\ln\left(5 x\right)\right)$.

### Find the first derivative $\frac{d}{dx} \left(\ln\left(5 x\right)\right)$

The function $\ln\left(5 x\right)$ is the composition $f{\left(g{\left(x \right)} \right)}$ of two functions $f{\left(u \right)} = \ln\left(u\right)$ and $g{\left(x \right)} = 5 x$.

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

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(5 x\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(5 x\right)\right)}$$

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

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(5 x\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(5 x\right)$$

$$\frac{\frac{d}{dx} \left(5 x\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(5 x\right)}{{\color{red}\left(5 x\right)}}$$

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 $c = 5$ and $f{\left(x \right)} = x$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(5 x\right)\right)}}{5 x} = \frac{{\color{red}\left(5 \frac{d}{dx} \left(x\right)\right)}}{5 x}$$

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, $\frac{d}{dx} \left(\ln\left(5 x\right)\right) = \frac{1}{x}$.

### Next, $\frac{d^{2}}{dx^{2}} \left(\ln\left(5 x\right)\right) = \frac{d}{dx} \left(\frac{1}{x}\right)$

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

$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)} = {\color{red}\left(- \frac{1}{x^{2}}\right)}$$

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

Therefore, $\frac{d^{2}}{dx^{2}} \left(\ln\left(5 x\right)\right) = - \frac{1}{x^{2}}$.

$\frac{d^{2}}{dx^{2}} \left(\ln\left(5 x\right)\right) = - \frac{1}{x^{2}}$A