Main Concept

The graph of the exponential function $f\left(x\right)\={\ⅇ}^x$ has a very interesting property. If you draw a vertical line (green in the graph to the right) from a point $\left(x_0\,{\ⅇ}^{x_0}\right)$ on the graph down to the point $\left(x_0\,0\right)$ on the x-axis, then draw another line 1 unit to the left (red), to the point $\left(x_0-1\,0\right)$, and then finally complete the triangle by drawing the line through $\left(x_0-1\,0\right)$ and $\left(x_0\,{\ⅇ}^{x_0}\right)$ (magenta), this final line will just touch the graph of ${\ⅇ}^x$ at this latter point without passing through the graph; that is, this line is tangent to the graph of ${\ⅇ}^x$ at the point $\left(x_0\,{\ⅇ}^{x_0}\right)$. This property (with the base of the triangle having length 1) and the specification that the function has value 1 at $x\=0$ completely and uniquely determine the exponential function ${\ⅇ}^x$.

The slope of this (magenta) line tells you how fast the function is growing. So this property of the exponential function can be summarized this way: At every point, how big the exponential function is and how fast it is growing are the same.

Click or drag on the graph to see a demonstration of this property.

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