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Derivatives of exponential functions.
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NooNoo studying calculus
a year to remember
Exponential growth rate ...
What is the derivative rule for exponential functions?
How are we going to remove that multiplier?
Where else can we find this constant? What does it represent?
Now let's look at this numerically ...
remember ... mathematics is the science of patterns
Mandala - the pattern of creation
Base c
Let's flip back and forth from the graph and look for a numerical patternthe search for the "perfect" number ...
Base c
Perfection, Colorado
Symbolically, we're looking for ...
a "magic" number, such that has slope 1 at x = 0.
Numerically ...
we now have 2 different definitions for the number e.
(1) e = the function whose slope of the tangent line to the graph of itself @ x = 0.(2)
So ...
But what about exponential functions with other bases ...
If If
then then
and
remember ... mathematics is the science of patterns
Mandala - the pattern of creation
Base c
Let's explore some more ...
How are all these values related? Let's graph them ...
Derivatives of Exponential Functions
The Exponential Function
any exponential function
Let's find these derivatives ...