Exponential FunctionEdit
First, we determine the derivative of using the definition of the derivative:
Then we apply some basic algebra with powers (specifically that a^{b + c} = a^{b} a^{c}):
Since e^{x} does not depend on h, it is constant as h goes to 0. Thus, we can use the limit rules to move it to the outside, leaving us with:
Now, the limit can be calculated by techniques we will learn later, for example Calculus/L'Hôpital's rule, and we will see that
so that we have proved the following rule:

Now that we have derived a specific case, let us extend things to the general case. Assuming that a is a positive real constant, we wish to calculate:
One of the oldest tricks in mathematics is to break a problem down into a form that we already know we can handle. Since we have already determined the derivative of e^{x}, we will attempt to rewrite a^{x} in that form.
Using that e^{ln(c)} = c and that ln(a^{b}) = b · ln(a), we find that:
Thus, we simply apply the chain rule:

Logarithm FunctionEdit
Closely related to the exponentiation is the logarithm. Just as with exponents, we will derive the equation for a specific case first (the natural log, where the base is e), and then work to generalize it for any logarithm.
First let us create a variable y such that:
It should be noted that what we want to find is the derivative of y or .
Next we will put both sides to the power of e in an attempt to remove the logarithm from the right hand side:
Now, applying the chain rule and the property of exponents we derived earlier, we take the derivative of both sides:
This leaves us with the derivative:
Substituting back our original equation of x = e^{y}, we find that:

If we wanted, we could go through that same process again for a generalized base, but it is easier just to use properties of logs and realize that:
Since 1 / ln(b) is a constant, we can just take it outside of the derivative:
Which leaves us with the generalized form of:

Logarithmic DifferentiationEdit
We can use the properties of the logarithm, particularly the natural log, to differentiate more difficult functions, such a products with many terms, quotients of composed functions, or functions with variable or function exponents. We do this by taking the natural logarithm of both sides, rearranging terms using the logarithm laws below, and then differentiating both sides implicitly, before multiplying through by y.

See the examples below.
 Example 1
Suppose we wished to differentiate We take the natural logarithm of both sides Differentiating implicitly, recalling the chain rule Multiplying by y, the original function 
 Example 2
Let us differentiate a function Taking the natural logarithm of left and right We then differentiate both sides, recalling the product and chain rules Multiplying by the original function y 
 Example 3
Take a function Then We then differentiate And finally multiply by y 