second derivative definition

Question

The definition of a derivative is $\frac{dy}{dx} = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} $ but while studying the wave equation I've encountered the following: $ tan(\theta_2) - tan (\theta_1)= \frac{\Delta \frac{dy}{dx} }{dx} = \frac{d^2y}{dx^2}$. Shouldn't $\frac{d^2y}{dx^2}$ be equal to $ \frac{ \frac{dy}{dx} }{dx}$ ?

Answer 1

By definition: $$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)$$ i.e. the symbol $\frac{d^2y}{dx^2}$ is the second derivative of $y$ with respect to $x$.

Here, instead of $\frac{d}{dx}$, the notation $\frac{\Delta}{dx}$ is used ($\Delta$ meaning "the change in").

Derivative operations are not fractions, though fractions can be a good analogy for some of their properties. So far as I can see, $\frac{ \frac{dy}{dx} }{dx}$ has no concrete meaning.