Derivatives Of Exponentials, Sine and Cosine
Derivatives of Exponential Functions
Recall that an exponential function is a function of the form
For concreteness to start out, we will look at the base
But using the laws of exponents,
It’s just the slope of
We can actually plot the function
The same thing works for
What does this mean about exponentials? The rate at which they grow is proportional to the value they already have! That means, the “per-capita” rate of increase is constant! This is why they show up in physical modeling all the time.
The importance of
When is this proportionality constant equal to
When
Derivative of Sine and Cosine
Use the angle sum rule for sin:
We can break this into the sum of two simpler pieces:
In the first of these we can factor out a
Just like for exponentials, we see that the two limits remaining do not depend on
The value of the second of these limits is directly giving the slope of
This appears to be 1 from the graph, which we can confirm by drawing
The other limit measures the slope of
Again we can confirm this by directly plotting the secant’s slope:
Becasue the first limit is 1 and the second is 0, this tells us that the derivative of
From this we can directly get the derivative of the cosine, recalling that
Thus,