Question

use the quotient rule

log x/8

Answer 1

[AI-generated answer — Bot GPT 3.5]

Let's use the quotient rule to differentiate the function:

The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then the derivative of f(x) is given by:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2

Now, let's apply the quotient rule to differentiate the function f(x) = log(x/8):

First, let's determine g(x) and h(x):

g(x) = log(x)
h(x) = 8

Next, let's find the derivatives of g(x) and h(x):

g'(x) = 1 / (x * ln(10))
h'(x) = 0 (since h(x) doesn't contain any x terms)

Now, we can apply the quotient rule:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
= ((1 / (x * ln(10))) * 8 - log(x) * 0) / (8^2)
= (8 / (x * ln(10)) / 64
= 8 / (64 * x * ln(10))
= 1 / (8 * x * ln(10))

Therefore, the derivative of f(x) = log(x/8) is f'(x) = 1 / (8 * x * ln(10)).