# Derivative of ex Proofs This function is unusual because it is the exact same as
its derivative. This means that for every x value, the slope at that point is equal
to the y value ## Limit Definition Proof of ex

Limit Definition: 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

Let Then 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

Let and consider 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.  Scroll to Top