# Lec 21 Fig 2: Exponential function and the slope of tangent

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.