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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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Your Input

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

Solution

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 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

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

Return to the old variable:

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

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

Answer

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