$\displaystyle P(t, x) = f(t - Ax*f(t))$ where $\displaystyle f(t) = P(t, 0)$

If one has f(t) = sin(t), then a shock occurs in P (infinite slope) for Ax = 1 (A is a constant).

I am trying to derive this result, i.e. I'm trying to show that when Ax = 1, dP(t, x)/dt has an infinite slope at some t.

$\displaystyle P(t, x) = f(t - Ax*f(t))$

$\displaystyle P(t, x) = f(u), u = t - Ax*f(t)$

I'm not sure how to proceed . Thanks for your input.