 Historically, calculus was created to solve three problems. First, given a region to find the area of the region. Next, given a function to find the greatest or least value of the function over an interval. And finally, given a curve, find a line tangent to the curve at a given point. Historically, the first problem, known as the quadrature or area problem, was the first to be solved. But for a variety of reasons, we typically teach calculus by solving the third problem. Now, before we try to find the tangent line, let's try and find what's known as the secant line. A secant line is a line that cuts Sicari and Latin through two points on a curve. Now, since we know how to write the equation of a line given two points, we can always write the equation of the secant line. For example, let y equals x squared plus 2x minus 7. Let's find a secant line through the point on the curve where x is equal to 1. Now, it's not necessary, but let's go ahead and put down a graph of our equation. Now, we need two points to write the equation of a line, and x equals 1 is a half point. We need the y value. Fortunately, we have a handy equation that tells us what the y value is. And so we find, and let's go ahead and write that down on our graph. Now, we'll need a second point, and since the problem doesn't limit us, we'll take, well, how about x equals 0 as our second half point, and we'll find the corresponding y coordinate, and label. So now we have two points, and we'll take our line between the two points, 1 negative 3 and 0 negative 7. So if we have a point on the line at the slope of the line, we can write down the equation in the point-slope form. And we have to find this slope. So remember the equation for finding the slope between two points. And so we can find the equation. We find the slope between the two points. And once we have the slope and a point, we can write the equation of the line. The line with slope 4 passing through 1 negative 3 has equation. So how do we solve the problem of finding a tangent to a curve? To find the equation of the tangent line to the curve y equals f of x, we'll need two things. First, a point a f of a on the curve, this is the point of tangency. And second, the slope of the tangent line. So how do we find that slope? Well, let's go back to the secant line. So remember the secant line runs between two points on the curve. So notice that if the two points of the secant line are close to the point of tangency, then the slope of the secant line approximates the slope of the tangent line. And this suggests the following approach. First, we'll find a second point on the curve b f of b that's close to the point of tangency a f of a. And then we'll find the slope of the secant line between a f of a and b f of b. And finally, we'll take the limit of the slope as the second point approaches the first. So let's find the equation of the tangent line to y equals x squared plus 2x minus 4 at x equals 3. It's extremely helpful to graph the equation, so let's graph it. Now we do need that point of tangency, and so at x equals 3, y equals, and so the point of tangency is 311. Let our second point be at x equal to b, then y is equal to b squared plus 2b minus 4, and the second point's coordinates will be b, b squared plus 2b minus 4. And that gives us a secant line. So now we have two points, and we can compute the slope of the secant line between the two points. And as our second point approaches the first, the slope of the secant line will approximate the slope of the tangent line. So that says we should take a limit. Now to find the limit, we note that at b equal to 3, both numerator and denominator are equal to zero, so both have a factor of b minus 3, and we find the other factor. And we can simplify, and we find the limit, and now we have a slope and the point on the line, and so we can write the equation of the line. Now to simplify the algebra, we can let our two points be a f of a, the point of tangency, and a nearby point, which we can think about as being a plus a little bit more, and the corresponding y value f of a plus a little bit more. Then the slope of the secant line has a nice form, and the limit of the slope of the secant line, as h goes to zero, will be the slope of the tangent line. And this is what we now call the derivative at a point. Now this formula might seem to be a bit mysterious, but all it really is is the limit of the slope of the secant line. And here's an important idea. If you only learn one thing in calculus, you probably failed the course. But among the things you should learn in calculus is that the derivative is the slope of the line tangent to the curve.