First, we determine the derivative of using the definition of the derivative:
Then we apply some basic algebra with powers (specifically that ab + c = ab ac):
Since ex 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 ex, we will attempt to rewrite ax in that form.
Using that eln(c) = c and that ln(ab) = b · ln(a), we find that:
Thus, we simply apply the chain rule:
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 = ey, 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:
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, re-arranging 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
We then differentiate
And finally multiply by y