We review differentiation and integration.
One of the fundamental objects of differential calculus is the derivative. As a reminder, here are some important results:
Powers of :
Inverse Trigonometric Functions:
Once the derivatives of basic functions have been established, we can use them to build the derivatives of more complicated functions:
- Scalar Multiplication:
- Product Rule:
- Quotient Rule:
- Chain Rule:
While there are many useful applications of derivatives, one of the most important ones is the relationship between derivatives and tangent lines.
Suppose that is a differentiable function on an interval . If we pick a value in the interval, and “zoom in” on the graph around , notice what happens:
As evidenced by the image, when the function is differentiable at a given -value, the graph of becomes closer to a line as we “zoom in,” and we call this line the tangent line at .
To find the equation of this line, we need a point of the line and the slope of the line. The slope of the line is and the point on the line is .
This result is also a direct consequence of the limit definition of the derivative and some algebra. If exists, then:
In order to write everything as a single limit, note we can write as:
We now return to the limit definition of and write:Denoting by , we note that is the equation of a line, and we can write the last equation above as:
Since this limit is , we can interpret it geometrically by saying that as the distance between and becomes smaller, the distance between the function and the line becomes even smaller. That is, as we “zoom in” on the graph of a function at a point where it is differentiable, the graph becomes closer and closer to a line and we call this line the tangent line.
As such, it is important to realize that a good intuitive way to think about tangent lines is that a function has a tangent line at a point if the graph of the function becomes less distinguishable from its tangent line as we continue to “zoom in” on the graph of the function at .