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We derive the derivative of the natural exponential function.
We don’t know anything about derivatives that allows us to compute the derivatives of
exponential functions without getting our hands dirty. Let’s do a little work with the
definition of the derivative:
There are two interesting things to note here: We are left with a limit that involves
but not , which means that whatever is, we know that it is a number, or in other
words, a constant. This means that has a remarkable property:
The derivative of an exponential function is a constant
Unfortunately it is beyond the scope of this text to compute the limit However, we
can look at some examples. Consider and :
While these tables don’t prove that we have a pattern, it turns out that Moreover, if
you do more examples, choosing other values for the base , you will find that the
limit varies directly with the value of : bigger , bigger limit; smaller , smaller limit.
As we can already see, some of these limits will be less than and some larger than .
Somewhere between and the limit will be exactly . This happens when We will
define the number by this property, see the next definition:
The number denoted by
, called Euler’s number, is defined to be the number satisfying the following
Using this definition, we see that the function has the following truly remarkable
The derivative of the natural exponential function The derivative of
the natural exponential function is the natural exponential function
itself. In other words,
From the limit definition of the derivative, write
Hence is its own derivative. In other words, the slope of the plot of is the same
as its height, or the same as its second coordinate. Said another way, the
function goes through the point and has slope at that point, no matter what
What is the slope of the tangent line to the function at ? The slope is .
Write with me:
We know the derivative of , but we’re being asked the derivative of . The
has been replaced by a function of . We think of as a composite function with
and . To take the derivative of a composite function, we have chain rule:
We know and which we calculate by product rule as (where we used chain rule
AGAIN inside the square root). Our answer is then: