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Mathematical Expression Editor
In this section we compute derivatives involving and .
The derivative of
We begin by computing the derivative of exponential functions of the form where .
To do this we will use the chain rule and a conversion formula to the natural base, .
The conversion formula is
Now, by the chain rule,
This gives us the following theorem:
Let , then
If , then since , we have which agrees with the formula we saw in section
2.3.
example 1 If then and if then .
(problem 1) Compute
Use the exponential rule with
The derivative of is
The derivative of is .
example 2 Find if Using the Sum Rule we have, .
(problem 2a) Compute
Use the Sum Rule
The derivative of with respect to is
(problem 2b) Compute
The derivative of is itself
The derivative of with respect to is
(problem 2c) Compute
Use the Constant Multiple Rule
The derivative of with respect to is
(problem 2d) Compute
Use the Product Rule with and .
.
The derivative of is
The derivative of is
The derivative of with respect to is
The Chain Rule applied to a general exponential function, , yields the following:
example 3 Find if We use the chain rule with as the inside function:
(problem 3) Compute
The derivative of
To find the derivative of with , we use the following change of base formula:
Since the denominator in the above formula is a constant, we can use the Constant
Multiple Rule to find the derivative:
This gives us the following theorem:
Let , then
If , then since , we have which agrees with the formula we saw in section
2.3.
example 4 The common log has base 10 and is usually written . So, if then