Use of differentiation operators?

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In Calculus, I am studying differentiation at the moment. The two equations is the basic Derivative function: (f(x+h)-f(x))/h and the alternative formula: (f(z)-f(x))/(z-x); and I can see how they both have their own purposes for finding the tangent line and such; but when will differentiation operators (dx/dy) ever be used practically in Calculus. Thomas' Calculus skims over it a little too quickly and doesn't flesh it out as much as I would have liked it to. Can someone help me out on how and why I would use it (compared to the other two methods) and if someone can back it up with an example, which is how I learn best, I would greatly appreciate it!

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The limits of both of those two difference quotients will give the derivative. In most calculus courses you will use them to derive a few simple derivatives, but then you will move on. They are used in developing derivative properties such as the product rule and quotient rule. Once you have a few basic formulas such as the ones for polynomials, trig functions, exponential and logarithm functions you will be able to differentiate most functions with the basic functions and rules such as the above rules and the chain rule. Nobody would try to differentiate$$
f(x) =\frac{\sqrt{1 + \tan x}}{e^{\sin 2x}}$$from its difference quotient.