Find the derivative of the function.
f(x)=ln(sqrt(x)+6)
To find the derivative of the function f(x) = ln(sqrt(x) + 6), we can apply the chain rule. The chain rule states that if we have a composition of functions, such as g(h(x)), then the derivative of this composition is given by g'(h(x)) * h'(x).
Let's break down the function f(x) into its composite functions:
g(x) = ln(x)
h(x) = sqrt(x) + 6
First, we'll find the derivative of g(x), which is g'(x):
g'(x) = (1 / x)
Next, we'll find the derivative of h(x), which is h'(x):
h'(x) = (1 / 2√x)
Now, we can use the chain rule to find the derivative of f(x):
f'(x) = g'(h(x)) * h'(x)
Substituting the functions we derived earlier:
f'(x) = (1 / h(x)) * h'(x)
f'(x) = (1 / sqrt(x) + 6) * (1 / 2√x)
Simplifying this expression:
f'(x) = 1 / (2 * √x * (sqrt(x) + 6))
Therefore, the derivative of the function f(x) = ln(sqrt(x) + 6) is f'(x) = 1 / (2 * √x * (sqrt(x) + 6)).