Question

Calculate using a chain rule:
y=In (sin x)

Answer 1

To differentiate the function \( y = \ln(\sin x) \) using the chain rule, we need to apply the chain rule to the composition of the functions involved.

1. Identify the outer and inner functions:
- Outer function: \( u = \ln(v) \) where \( v = \sin x \)
- Inner function: \( v = \sin x \)

2. Differentiate the outer function with respect to the inner function \( v \):
\[
\frac{du}{dv} = \frac{1}{v} = \frac{1}{\sin x}
\]

3. Differentiate the inner function with respect to \( x \):
\[
\frac{dv}{dx} = \cos x
\]

4. Now apply the chain rule:
\[
\frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx}
\]

5. Substitute the derivatives calculated:
\[
\frac{dy}{dx} = \frac{1}{\sin x} \cdot \cos x
\]

6. Therefore, the final result, simplifying the product, is:
\[
\frac{dy}{dx} = \frac{\cos x}{\sin x} = \cot x
\]

So the derivative of \( y = \ln(\sin x) \) is:
\[
\frac{dy}{dx} = \cot x
\]