In differential calculus, our main goal is to find the instantaneous rate of change of some curve at any given point. To better understand this, let’s think back to linear functions, and the idea of slope.
The graph shown to the above is . Recall that the equation of a line gives you two pieces of information, the slope of the line, and the y intercept. With this equation, we can see that the slope is 2, and the y intercept is 0. This is also clear from the graph of the equation.
With linear functions, it is easy to figure out how much the graph is changing at any given point. Since the slope is constant, 2, we know that at any given point, the graph has a rate of change of 2. Differential calculus allows us to extend this idea to functions that do not have a constant slope. For example, consider the graph below.
This graph shows the function ,which is a quadratic function. With quadratics, the rate of change varies on each point. With our current math techniques, we don’t have a good way to describe the way that quadratics change at any given point. This problem is not limited to quadratics, for that matter, more functions have variable slopes than constant. Due to the non linearity of the world, calculus becomes a very important mathematical skill.
So, let’s get some intuition about how we might be able to solve the problem of finding the instantaneous rate of change of the function. Understanding what we need to do will help us understand the motivation behind each section of calculus. We know the slope of a linear function can easily be found, using a formula like . If we can draw a line that touches a point on a function, or is tangent to the function, we would be able to know the slope at that given point. To demonstrate, look at the graph below.
In this graph, the red line is and the blue line is . You can see that the blue line is tangent to the red line, meaning it touches the red line in one place, x = 2. Since it is tangent, the slope of the blue line will tell us the rate of change of the red line at the point x = 2.
So, how did we figure out that the blue line is tangent to the red line? Consider the equation of a slope, . This equation requires two points in order to be solved. If we make the second point as close as possible to the first point, we can get a difference so small that they are basically the same point. In calculus, we refer to this idea as a limit, and it is the first concept we will learn in calculus. Once we have the ability to compute limits, we can compute the slope of the tangent line. From here, we can simply substitute in our point for , and find the tangent line.