Derivative of ln(x)^2

The calculator will find the derivative of

\ln^{2}\left(x\right)

, with steps shown.

Your Input

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

Solution

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

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

Apply the power rule $\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$ with $n = 2$ :

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

Return to the old variable:

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

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

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

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

Answer

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

Derivative of ln⁡2(x)\ln^{2}\left(x\right)ln2(x)

Your Input

Solution

Answer

Derivative of $\ln^{2}\left(x\right)$