Figure 1.15: Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A.
This section provides a quick introduction to the idea of the derivative. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation.
Often we are interested in the slope of a line tangent to a function at some value of . This slope is called the derivative and is denoted . Since a tangent line to the function can be defined at any point , the derivative itself is a function of :
As figure 1.15 illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve:
As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.