Find the derivative of the function.
f(x) = 6 (ln(x))^(5\/2)
To find the derivative of the function f(x) = 6(ln(x))^(5/2), we can apply 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).
Let's break down the function and apply the chain rule step by step:
Step 1: Identify the inner function, g(x).
In this case, the inner function is ln(x).
Step 2: Find the derivative of the inner function, g'(x).
The derivative of ln(x) is 1/x.
Step 3: Determine the outer function, f(g(x)).
In this case, the outer function is (ln(x))^(5/2).
Step 4: Find the derivative of the outer function, f'(g(x)).
To find the derivative of (ln(x))^(5/2), we can use the power rule. The power rule states that if we have a function of the form f(x) = x^n, then its derivative is given by f'(x) = nx^(n-1).
Applying the power rule to (ln(x))^(5/2), we get:
f'(g(x)) = (5/2)(ln(x))^(5/2 - 1) * (1/x)
= (5/2)(ln(x))^(3/2) * (1/x)
Finally, combining the results from step 2 and step 4, we can find the derivative of f(x):
f'(x) = f'(g(x)) * g'(x)
= (5/2)(ln(x))^(3/2) * (1/x)
Therefore, the derivative of the function f(x) = 6(ln(x))^(5/2) is:
f'(x) = (5/2)(ln(x))^(3/2) / x