In the following graph, there are two lines: f(x) which is black and f(y) which is blue. f(y) is the inverse of f(x) over the line y=x. Their tangents, which are red and green, show the slopes at that instantaneous point on the graph. As you move the point x, the different slopes can be seen. Since we know that the tangent line is the instantaneous slope at a point, and we know that instantaneous slope is the same as the derivative. We can conclude that the derivative is equal to the tangent's slope. Because of this conclusion, the value is labeled with f'(x).
| f(x) | ||||
| f'(x) | ||||
| 1/f'(x) | ||||