How can a limit become second derivative?

Question

I'm physics student and studying waves. I don't know the mathematical background of this formula.

$\dfrac{\partial u(x+\Delta x,t)}{\partial x} - \dfrac{\partial u(x,t)}{\partial x}\approx\Delta x\dfrac{\partial^2 u(x,t)}{\partial x^2}$

I suspect it has to do with Taylor series, but how can a limit become second derivative?

Answer 1

This is actually a standard first derivative, except the function whose derivative is being taken is itself already a first derivative of another function. So the derivatives 'pile up', so to speak.