Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem

Find derivative for $lnx^2$.

It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha.

I don't understand how to calculate the derivative for $lnx^2$, and the differents between these two expressions.

Could anyone clear this matter?

Any help will be appreciated.

1 Answer

Here $\ln(x)^2 = (\ln(x))^2 = \ln^2(x)$ where you are taking $\ln$ of $x$ and then squaring, where as in $\ln(x^2)$ you square the $x$ first and then take $\ln$. Do you see the difference?

The derivative of the first (using the chain rule or product rule) will be:

$$\frac{\partial}{\partial x}(ln(x))^2 = 2\ln(x)\frac{1}{x} = \frac{2\ln(x)}{x}$$

The derivative of the second will be:

$$\frac{\partial}{\partial x}(ln(x^2)) = \frac{1}{x^2}2x = \frac{2}{x}$$

You could also notice that $\ln(x^2) = 2\ln(x)$ and take the derivate from there.