Question

If f(x)=ln (7x+ln (x)) , find f'(1). f'(1)=

A. 1/1
B. 1/7
C. 1/8
D. 1/14

Answer 1

To find f'(1), use the chain rule to differentiate f(x) = ln(7x + ln(x)). The derivative is 1/7.

Step-by-step explanation:

To find the derivative of f(x) = ln(7x + ln(x)), we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative is given by f'(g(x)) * g'(x). In this case, g(x) = 7x + ln(x) and f(x) = ln(g(x)).

First, find the derivative of g(x) = 7x + ln(x). The derivative of 7x is 7, and the derivative of ln(x) is 1/x. So, g'(x) = 7 + 1/x.

Now, find the derivative of f(x) using the chain rule. f'(x) = (1/g(x)) * g'(x). Substitute g(x) and g'(x) into the equation to get f'(x) = (1/(7x + ln(x))) * (7 + 1/x).

To find f'(1), substitute x = 1 into the equation. f'(1) = (1/(7(1) + ln(1))) * (7 + 1/1) = 1/7.