 Let's take a look at a few more derivatives. So, let's find the derivative of f of x equals square root of x plus 5 at x equal to 4. Now, again, remember that the derivative is the same as the slope of the line tangent to the graph of, and I could sketch a graph of y equals f of x, y equals square root of x plus 5, but let's do something a little bit different for variety. Let's construct a table of values for my function. So, what do I need to know? Well, at x equals 4, then I have a function value square root of 4 plus 5, that's square root of 9, that's 3, and so my function value at 4 is 3. And for the derivative, I need to know the function value at 4 plus h. So, at x equal to 4 plus h, I'll substitute that into my function, and after all the dust settles, my function value is square root of 9 plus h. And now I can find the derivative. So, all I'm doing here is I'm just copying down the definition of the derivative, that the derivative of f at 4 is the limit as h goes to 0 of f of 4 plus h minus f of 4 over 8. And the nice thing about the table organization is that I can just look up these values. f of 4 plus h, well, here it is. f of 4, there it is, and I can just substitute them in as my initial expression. And as h goes to 0, numerator goes to square root 9 minus 3, that's 0 over 0. This is an indeterminate form, so I have to do something. In this case, I'll multiply by the radical conjugate, simplify, simplify, simplify, and now I can evaluate this limit as h goes to 0. So as h goes to 0, 1 stays 1, 3 stays 3, and square root of 9 plus h goes to square root of 9. Square root of 9 is just 3, so this is 1 over 3 plus 3, and that simplifies to 1 over 6. Well, what if I change the x value? So I'll keep the function, square root of x plus 5, but maybe I want to find the derivative that, oh, I don't know, x equal to 1. I can do exactly the same thing, so I'll construct a table, and my function values, I want to find the function value at 1, there it is. I want to find the function value at 1 plus h, 1 plus h, substituting it in, there it is, and my derivative again is just defined by the limit as h goes to 0 of a particular expression, and so I can substitute in my values. So I know what f of 1 plus h is, I know what f of 1 is, and there's my function expression. Again, it's a radical expression as h goes to 0, numerator goes to 0, denominator goes to 0, it's an indeterminate form, and so I do some algebra, do some more algebra, do some more algebra, do some more algebra until I get an expression that I can evaluate. As h goes to 0, this expression goes to 1 over square root of 6 plus square root of 6. And well, I can do a little bit of simplification there, that's 1 over 2 radical 6. Well, here's the heart and soul of mathematics. I could go through all of this work every time I wanted to find a derivative, but the heart and soul of mathematics is generalization, and there's two layers to that generalization. That first layer is given a function, given a particular function, rather than finding the derivative at some specific given point, x equal to a, I might try to find a formula for the derivative of the function at any point that I want. So I'm looking for a general derivative formula for the function, and so there's the first generalization. And then, I might not want to have to go through that algebra every single time, so maybe what I can do is find a formula for the derivative of any function at any point. And these two questions will take up the next couple of weeks, so first off, we'll look for a formula for the derivative of a specific function at any point, and then we'll see if we can generalize that process so we can find the derivative of any function.