This means "find the instantaneous rate of change of f at 1" or "find the derivative of f at 1." We use the limit definition of the derivative to find

Example 2

If y = x^{2}, what is when x = 1.

This is using Lagrange instead of Leibniz notation. We're asked to find the instantaneous rate of change of y with respect to x when x = 1. This is the same as the derivative of y with respect to x when x = 1. So we use the limit definition of the derivative again:

We've already said this notation is nice because it reminds us we're talking about slopes. It also allows us to specify variables more carefully. When we start talking about multiple variables that relate to each other, we might want to talk about quantities such as the rate of change of x with respect to y. It's not immediately clear how we would do this with Lagrange notation, but with Leibniz notation, it's do-able. The instantaneous rate of change of x with respect to y when y = b is

We could also write the instantaneous rate of change of x with respect to y when x = a:

We can also translate notation back into words. We find the instantaneous rate of change of (the variable on the top) with respect to (the variableon the bottom). The expression

means "the instantaneous rate of change of K with respect to R when R = 2."