Derivative of ln(sin(x))

Step-by-step solution

1. 1Chain Rule — ln
   $$\frac{d}{dx}\left[\ln \left(\sin \left(x\right)\right)\right]=\frac{1}{\sin \left(x\right)}\cdot \frac{d}{dx}\left[\sin \left(x\right)\right]$$

   The derivative of $\ln (u)$ is $\frac{1}{u}$. Here $u=\sin \left(x\right)$, so the outer part is $\frac{1}{\sin \left(x\right)}$. Multiply by $u^{\prime }$ (chain rule).

2. ↳Find u′: Differentiate the inner function, $u=\sin \left(x\right)$:
3. 2Derivative of sin
   $$\frac{d}{dx}\left[\sin \left(x\right)\right]=\cos \left(x\right)$$

   The derivative of $\sin (x)$ is $\cos (x)$ — a standard rule.

4. =Combine
   $$\frac{d}{dx}\left[\ln \left(\sin \left(x\right)\right)\right]=\frac{1}{\sin \left(x\right)}\cdot \cos \left(x\right)=\frac{\cos \left(x\right)}{\sin \left(x\right)}$$

   With $u^{\prime }=\cos \left(x\right)$, multiply by the outer derivative to complete the chain rule.

5. ✓Final Answer
   $$\frac{d}{dx}\left[\ln \left(\sin \left(x\right)\right)\right]=\frac{\cos \left(x\right)}{\sin \left(x\right)}$$

   The first derivative simplifies to $\frac{\cos \left(x\right)}{\sin \left(x\right)}$.