Consider the exponential function \[y = b^x.\] We know that if a quantity \(y\) is exponential in a variable \(x\), the rate of change of \(y\) per unit \(x\) is proportional to its value \(y\). We therefore expect that the slope of its tangent line is a constant multiple of \(y\), i.e. \[\text{(slope of \(y\))} = ny.\] It is natural to ask if there exists a value \(b\) for which the function \(y = b^x\) has the same value as the slope of its tangent line at \(x\), i.e. we want to find \(b\) which produces \(n = 1\).
The following graph shows both the function \(y = b^x\) and its slope function. If you set \(b = 2.7\), you will notice that both functions almost overlap each other (i.e. \(n \approx 1\) when \(b = 2.7\)). The exact value of \(b\) that produces \(n = 1\) is denoted by the symbol \(e\), which has the value \[e = 2.7182818284590...\]
You may change the values of \(b\) and \(x\) in the following graph to see how both function change.