I'm developing a mathematical model for a physical system and have come across the following logical quandary (at least, it is for me). I'm having trouble proving whether the following proposition is true or false. Any input on its truth or falsehood would be much appreciated. Also if its truth or falsehood is easily, formally proved, I would be very interested in the proof itself:
Let $x$ be a real, non-negative number and $y(x)$ also be a real, non-negative, univariate, scalar function of $x$. If:
\begin{equation} y(x) \leq x \,; \end{equation}
for all valid $x$ and $y$, then:
\begin{equation} \frac{d}{dx}y(x) \leq \frac{d}{dx}x \,; \end{equation}
or in other words:
\begin{equation} \frac{dy(x)}{dx} \leq 1 \,; \end{equation}
for all valid $x$ and $y(x)$.
Thank you!
It is certainly not true. $\frac{dy}{dx}$ is the gradient of $y$, and just because $y$ is less than $x$ doesn't mean it has to have a gradient less than $x$. A counterexample is the function $y(x)=e^{x-2}$ defined on the window where it is less than $x$, which contains the point $x=3$. But $y'(3)=e>1$.