To differentiate the function f(x) = sin(3 ln(x)), we will use the chain rule, which is essential when dealing with composite functions.
The chain rule states that if you have a composite function f(g(x)), then its derivative is given by f'(g(x)) imes g'(x).
In our case, we can identify the outer function as sin(u), where u = 3 ln(x). First, we need to find the derivative of the outer function:
- Derivative of sin(u): The derivative of sin(u) with respect to u is cos(u).
Next, we find the derivative of the inner function:
- Derivative of u = 3 ln(x): The derivative of ln(x) is 1/x, so by applying the constant multiple rule, we have:
- u’ = 3 * (1/x) = 3/x.
Now, applying the chain rule:
- f'(x) = cos(3 ln(x)) * (3/x)
This simplifies to:
- f'(x) = (3 cos(3 ln(x))) / x.
So, the derivative of the function f(x) = sin(3 ln(x)) is f'(x) = (3 cos(3 ln(x))) / x.