Step 1:

We notice that the given function is a composite function, or a function within another function. To find the derivative of composite functions, we will use the chain rule. We can see that the inner function is [latex]\ln(x)[/latex] and the outer function is the cosine function.

The chain rule states:

[latex]\Large{\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)}[/latex]

Step 2:

Applying the chain rule, we have:

[latex]\large{\frac{d}{dx}\cos(\ln(x))}[/latex]

[latex]\large{ = -\sin(\ln(x)) \cdot \frac{d}{dx}(\ln(x))}[/latex]

Step 3:

Simplifying, we have:

[latex]\Large{ = -\sin(\ln(x)) \cdot \frac{1}{x}}[/latex]

[latex]\Large{ = \frac{\sin(\ln(x))}{x}}[/latex]

Thus, this is the final answer for the derivative. So:

[latex]\Large{g'(x) = \frac{\sin(\ln(x))}{x}}[/latex]

Solution Complete!