Search 74,141 tutors

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



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


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.

Sign up for free to access more calculus resources like . WyzAnt Resources features blogs, videos, lessons, and more about calculus and over 250 other subjects. Stop struggling and start learning today with thousands of free resources!