Derivative of ex Proofs
This function is very unique. It is the only function that is the exact same as it's derivative. This means that for every x value, the slope at that point is equal to the y value
Limit Definition Proof of ex
By laws of exponents, we can split the addition of exponents into multiplication of the same base
Factor out an ex
We can put the ex in front of the limit
We see that as h approaches 0, the limit will get closer to 0/0 which is an indeterminant form (meaning we don't really know what is happening to to value as both the numerator and denominator approach 0). What we can do is plug in the point (0,1) and see the function's behavior at that point.
This limit definition states that e is the unique positive number for which
which we can clearly see on the graph.
Using this defition, we can substitute 1 for the limit
Implicit Differentiation Proof of ex
Taking the derivative of x and taking the derivative of y with respect to x yields
Multiply both sides by y and substitute ex for y.
Proof of ex by Chain Rule and Derivative of the Natural Log
From Chain Rule, we get
We know from the derivative of natural log, that
We also know that ln(e) is 1
Now we can substitute 1 and 1/u into our equation
Multiply both sides by u
and substitute ex for u.