 So let's see what else we can do with the derivative. The first important observation is the actual value of A doesn't affect how we find the derivative. And so this means we can generalize our definition to the derivative and go from the derivative at a point to the idea that the derivative is a function. So let f of x be a function. The derivative of f with respect to x, written f prime of x, is the limit as h goes to zero of the difference quotient, where instead of letting x be equal to a, we're just going to leave x as x. So here's a good rule of thumb in mathematics. And in life in general, the more ways we have of writing about something, the more important that thing probably is. And this is because many people have identified something as important and have invented their own way of expressing or talking about it, and only later did we realize that we were all talking about the same thing. And in this particular case, the notation for the derivative has three common forms. So what we've introduced is called prime notation. The derivative of f of x is written as f prime of x. And prime notation has the advantage of being nice and compact and easy to write. Unfortunately, it's not very informative. And as a result, one of the other forms of notation is differential notation, where we indicate the derivative of f with respect to x is written as df over dx. Now, this looks like a fraction, and in many ways we're going to pretend that it's a fraction, but the important thing here is that it's not actually a fraction. We'll go into detail into some of the important differences a little bit later on. What is important to identify here is that our differential notation tells us not only what function we're dealing with, but also the variable that we're differentiating with respect to. And finally, the third common way of expressing the derivative is to use what's known as subscript notation. This derivative is written as f subscript x. Prime notation and differential notation are the most common. Subscript notation typically shows up much later in the mathematics sequence. So for example, we might use the definition to find dy dx for y equals square root of x. So remember this dy dx is the same as f prime of x, where we know what our function is. So we'll pull in our definition of the derivative, and so we need to evaluate our function at x plus h and at x. So one of the things we can do to make this a little bit easier is to drop out our variable and replace it with an empty set of parentheses. So I want to know y at x plus h, so I'll drop in an x plus h. And I want to know y at x, so I'll drop in an x. And I'll substitute these values into my definition of the derivative. Now, we can quickly verify that at h equal to zero, both numerator and denominator are zero, so we have an indeterminate form, which means we typically can do something algebraically. And in this case, since we have a square root, let's multiply by the conjugate. So the conjugate will have the same terms, except there will be a plus instead of a minus or a minus instead of a plus. So our conjugate, square root of x plus h plus square root of x, and since we don't want to change the value of our expression, we need to divide by square root of x plus h plus square root of h. So the problematic thing is this square root of x plus h minus square root of x. So we'll multiply out our numerator. On the other hand, the denominator is less of a problem and we like it in factored form, so let's leave it as h times square root of x plus h plus square root of x. Multiplying out the numerator and doing some algebraic simplification, and we try again. If h actually is 0, our numerator is 1 and our denominator is not 0, so we can evaluate this and so our limit is going to be 1 over 2 square root of x. Well, let's try another one. So again, we know that we need to use the definition to find the derivative. And so we want to find y at x and at x plus h. So again, our function is y equals 1 over x, so I'll drop out the x and leave an empty set of parentheses. At x plus h, y is equal to 1 over x plus h, and at x, y is equal to 1 over x. So I'll substitute those back into our definition of the derivative and we'll do a little bit of algebra. Well, actually we'll do a lot of algebra. So again, it's worth checking that at h equal to 0, the numerator and denominator are both going to be 0, so we have an indeterminate form, which means we have the possibility of doing some algebra to simplify. One of the things we can do here is if we multiply through by our denominator x times x plus h, we'll be able to get rid of these compound fractions. And again, factored form is nice. The denominator h times x times x plus h is something we might be able to work with. The numerator on the other hand, let's multiply that out and do a little bit of algebra and a little bit more algebra. And finally, at the very, very, very, very, very end of our process, we are ready to do some calculus. So now we can take the limit as h approaches 0 of this expression and again, we can check at h equal to 0, this is a definite form of minus 1 over x squared.