Part 3/7:
To illustrate the concept geometrically, consider a curve represented by ( y = f(x) ). We can select a point ( P ) at ( (x, f(x)) ) and shift horizontally to ( Q ) at ( (x + \Delta x, f(x + \Delta x)) ). The change in ( y ) as we move from point ( P ) to point ( Q ) is denoted as ( \Delta y ).
The derivative provides a linear approximation of this curve at point ( P ) by using the tangent line. The slope of this tangent line at ( P ) encapsulates the notion of ( dy ). Thus, if we draw a tangent line, we can express the change in ( y ) through differentials accurately.
For the point ( P ):
- Coordinates: ( (x, f(x)) )
For point ( Q ):
- Coordinates: ( (x + \Delta x, f(x + \Delta x)) )