Question

In class I gave a general formula for derivative of an exponential function f(x) = log_a u(x); where "a" is called Base, and u(x) is called the argument of the log function, which is also given below.

Note, the derivative of a logarithmic function f(x) has three parts and is given here:

f ' (x) = 1/ln (a) .   1/u(x) . u ' (x)

In your words, describe (in words only)  each one of the three parts shown above:

a- describe part one

b- describe part two

c- describe part three.

Answer 1

We know that loga u(x) can be written as :

logau(x) = (ln u(x)) / (ln a) = p(x)

Now, when we differentiate a function that has another function in it, we use the chain rule. In the problem given, chain rule has been used. Since (1/ln a) is a constant multiplied to ln u(x), when we differentiate p(x), it remains the same. That is part one. Then according to the chain rule, we differentiate ln u(x). Since differential of the function lnx is 1/x, we get part two. Since u(x) is also a function of x, according to chain rule, it also needs to be differentiated and multiplied to part one and two to get the final differential. The differential of u(x) is u'(x) and that is part three.