What is the derivative of ln(sin(x))?

Steps to Solve

1. Identify the function to differentiate

We need to differentiate the function $y = \ln(\sin(x))$.

2. Apply the chain rule

To differentiate this, we use the chain rule, which states that if you have a composite function $y = f(g(x))$, then the derivative $y'$ is given by

$$ y' = f'(g(x)) \cdot g'(x) $$

Here, $f(g) = \ln(g)$ where $g(x) = \sin(x)$.

3. Differentiate the outer function

First, we differentiate the outer function $f(g) = \ln(g)$.

The derivative of $\ln(g)$ is

$$ f'(g) = \frac{1}{g} $$

4. Differentiate the inner function

Next, we differentiate the inner function $g(x) = \sin(x)$.

The derivative of $\sin(x)$ is

$$ g'(x) = \cos(x) $$

5. Combine the derivatives

Now we combine the results from steps 3 and 4 using the chain rule:

$$ y' = f'(g) \cdot g'(x) $$

Substituting $g = \sin(x)$:

$$ y' = \frac{1}{\sin(x)} \cdot \cos(x) $$

This simplifies to:

$$ y' = \frac{\cos(x)}{\sin(x)} $$

6. Final simplification

We recognize that $\frac{\cos(x)}{\sin(x)}$ can be simplified further to:

$$ y' = \cot(x) $$