# Finding a tangent using a point that is undefined for the function

$f(x) = x\ln(a^2x^2), a > 0$

A tangent to the derivative of the function goes through $(0, 0)$.

The task is find the tangent's intersection point with the derivative and the function of the tangent.

So I derived $f(x)$ twice and got to $f''(x) = {2\over x}$.

How do I continue from here?

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Let $(p,f'(p))$ be the point of tangency. (You calculated $f'(x)$ on your way to $f''(x)$.)
Set up the equation of the tangent line at the unknown point $(p,f'(p))$. It is $$\frac{y-f'(p)}{x-p}=\frac{2}{p}.$$
This line goes through $(0,0)$. That will give you an equation for $p$, which is easily solved.