 For real variables, we define the derivative as... Unfortunately, for complex functions, we have to introduce a complex definition for the derivative, which will be... Where the complex part is here. For example, let f of z equals z squared. Let's find f prime of z. So, definitions are the whole of mathematics. All else is commentary, so we'll pull in our definition of the derivative. And so we find... So, in calculus, you always found the derivative using the definition for about a week. Then you use the derivative rules for most functions. The derivative rules, since they're based on limit theorems, also hold for complex functions. So, for example, let f of z equals z squared plus e to the z, find f prime of 2 plus i. So, from calculus, we know the derivative of 2x is 2, and the derivative of e to the x is e to the x. These are still true for complex numbers, so we find that f prime of z is... And evaluating that at 2 plus i gives us... And we can do that for any function. So, suppose f of z is the real part of z, we'll find f prime of 3. So, from calculus, we know the derivative of the real part of x. Well, actually, we don't. We didn't run into this function in calculus. Well, that's okay. We can always go back to the definition. So, using the definition, we find f prime of 3 will be... We need to fill in some of the pieces, so the real part of 3 plus h will be... And the real part of 3 is just 3. And so our difference quotient becomes... And we need to know what happens to the real part of h as h goes to 0. So, remember, the limit must be the same regardless of how we approach it. So, suppose h is a real number, then the real part of h is h, and our limit will be... So, if h is a real number, then the limit is 1. But if h is a pure imaginary, the real part of h is 0, and our limit will be... So, if h is a pure imaginary number, the limit is 0. And since the values don't agree, the limit doesn't exist. And since the derivative is the limit, the derivative doesn't exist. And this is one of the big differences between complex analysis and real analysis. In real analysis, a continuous function with no derivatives anywhere takes a lot of effort to create. And outside of a real analysis class, you probably won't see them. In complex analysis, continuous functions with no derivatives anywhere are easy to find. The real part of c, the imaginary part of c, and a lot of other functions. So how can we tell if a complex function has a derivative without invoking the limit definition? We'll take a look at that next.