To express the relationship between $\Delta y$ and $\Delta x$ in terms of $f(x)$, we can use the difference quotient of $f(x)$ given by the formula $ \frac{f(x+h)-f(x)}{h} $ where $h$, or $\Delta x$, represents the difference between $x$ and another sampled value along the $x$ axis. Essentially, we're subtracting $y$, or $f(x)$, from $y + \Delta y$, $\Delta y$ being the difference between $f(x+h)$ and $f(x)$, meaning $ \frac{f(x+h)-f(x)}{h} = \frac{\Delta y}{\Delta x} $ For instance, given a function $f(x) = x^2 + 1$, then, following the rules of [[fieldnotes/Maths/Algebra Notes/Functions#^function-composition|function composition]], we can infer that $ f(x+h)=x^2+h^2+2xh+1 $ Therefore, $ \frac{f(x+h)-f(x)}{h} = \frac{(x^2+h^2+2xh+1)-(x^2+1)}{h} = h + 2x $ The difference quotient is therefore $h + 2x$. With the difference quotient, we are now able to compute the slope of a curve at any given point on the function graph as $h$ approaches $0$. To do so, we can substitute $h$ with $0$, meaning we're computing the slope at an *instant*. This slope gives us the direction in which we need to go to find the next point on the graph, which requires of us to think of a curve or a line as an infinite collection of points. I find the [[Cantor Set]] to be a good proxy to visually represent this idea. From our previous example, this gives us a slope of $2x$ at any given point (instant) on the graph.