The development of Calculus
Slope of a Curve: The graph of a linear function is always a straight line. That means between any two points on the line (if we connect them), the slant or better known as Slope between any two points is always the same. Slope is defined as the change in y-values divided by the change in x-values between any two points. For a staight line this is always the same. Take for example our line y=2x. If we take any two points, say, (1,2) and (3,6) and calculate the y change it would be 6-2 or 4. The x change between the same two points is 3-1 or 2. Dividing the y change by the x change we ge 4/2 or 2. If we did this many times changing the points we would always get the same result 2. We can say that the slope of this line is 2. It is visible in the equation in front of the x. This is an important feature of the equation of a staight line in the form of y=mx+b that most of you were exposed to in lower grades. The popular symbol for slope is m.
However, if our curve is not a straight line, then the curve will not have the same slope between any two points. We must rethink what we mean for the slope of the curve.
In those cases, we use a tangent line (touches the curve in exactly one point) and concentrate how we can get the slope of that tangent line. This will lead us into the most important concept of Calculus. That is the concept of a "LIMIT".
This concept is the foundation of Calculus. For the serious math student, much time is spent here and much analysis is given in a formal course. This is one concept students often find very difficult to comprehend since it involves much theory and many formal proofs. Very difficult for beginning calculus students. No worries, I will skip all of that and try to give you a very informal view.
Slope of the Secant line: A secant line is a line connecting two points on a curve. If the curve is not a straight line, it will only intersect the curve at those points in the neighborhood of those points. The rest of the secant line is off the curve. If we "nail" down one of those two points (keep it stationary) and let the other point move closer to that stationary point, the x and y change between those points become smaller. The closer we allow the movable point on the curve to get to the stationary point, the smaller & smaller the the x and y changes become. As that point moves closer to the stationary point secant lines lines are drawn, we will notice that these secant lines get closer and closer to the tangent line drawn at the stationary point. Using math language, we say that as the movable point approaches the stationary point, the secant lines approach the tangent line. So, that would mean that the slopes of these secant lines will approach the slope of the tangent line (this is what we are after). By formulating an expression that represents the slopes of the secant lines all we have to do is to let the change in x approach 0 and see if this expression gets close to a value. If it does, we have it. Easier said then done.
From our previous discussion, we know that the slope of any line between two points is defined by the change in y over the change in x. Writing this in mathematical symbols,
this is represented by Δy/Δx. The Δ symbol reads "change in".
As the movable point on the curve approaches (symbol for approaches is →) the stationary point Δx →0 and the slopes of the secant lines → the slope of the tangent line.
Since this slope is so important to us, we give it a special symbol and a special name. The symbol is dy/dx and we call it the Derivative of the function at this point. So,
your first interpretation (and most basic) of the derivative of a function is an expression that will give the slope of the tangent line to the graph of the function at any point you desire, just substitute the x value of the point into the derivative expression to get it. There will be other important interpretations that will be be given as we continue our discussion. Getting the actual expression for the derivative may take effort and a good knowlege of algebra, but is presented carefully in a formal course.