Here are two forms of shorthand we will occasionally allow ourselves, to simplify the notation. At the top, if it is clear that the result of f is a function of x, we will omit its argument. You can think of this in Leibniz's terms if you like: we divide the change in f by the change in x. What is left more implicit by this notation is that we create the change in x, and observe the resulting change in f.
The second line shows what happens when we fill in the explicit functional form of f in terms of x. We will do this very often. Technically, we need to put brackets around the whole function, or the statement might be ambiguous (especially if we write the function to the right of the division line as in the previous slide). In practice, things look a lot clearer without the brackets, so if the potential ambiguity is minimal, or can easily be resolved from context, we allow ourselves to leave the brackets out.
+Here are some examples. The first two lines show the derivatives we've already worked out in the Lagrange notation.
The right part of the second line shows how much clearer things can become when we assume that we know which variable is dependent on which. The notation is more ambiguous, but a lot clearer.
The third line shows the benefit of indicating the independent variable. The variables a, b, c are indicated with letters in the function, but we treat them as constants: x is the only variable we change to observe the resulting change in the function above the line. The rest is treated the same way as the 3 in the exponent is.