Integral Calculus, Exponential Function
Exponential Function
Since log(x) is a 1-1 function from x > 0 onto y,
we can define an inverse function
from x onto y > 0.
Simply reflect the log curve through the main diagonal.
This is the exponential function.
As x becomes negative the curve flattens out against the x axis,
just as the log curve approached the y axis.
As x becomes positive the exponential function rises to infinity.
The exponential of x is written exp(x), or sometimes Ex.
We know that log(exp(x)) = x for any x.
Take derivatives and apply the
chain rule.
The derivative of exp(x), over exp(x), gives 1,
hence the derivative of exp(x) is itself.
At every point, the function equals its slope.
The nth derivative is the same as the first, namely exp(x).