คำตอบ - Derivative

Computing the derivative of a logarithm function using the chain rule.

$$\frac{d}{dx}\left[\ln \left(\sin \left(x\right)\right)\right]=\frac{1}{\sin \left(x\right)}\times \frac{d}{dx}\left[\sin \left(x\right)\right]$$

Decomposing the function for the chain rule.

$$\frac{d}{dx}\left[\ln \left(\sin \left(x\right)\right)\right]=\frac{d}{dx}\left[\ln \left(x\right)\right]\times \frac{d}{dx}\left[\sin \left(x\right)\right]$$

Computing the derivative of a natural logarithm function.

$$\frac{d}{dx}\left[\ln \left(x\right)\right]\times \frac{d}{dx}\left[\sin \left(x\right)\right]=\frac{1}{x}\times \frac{d}{dx}\left[\sin \left(x\right)\right]$$

Substituting the variable back into the function.

$$\frac{1}{x}\times \frac{d}{dx}\left[\sin \left(x\right)\right]=\frac{1}{\sin \left(x\right)}\times \frac{d}{dx}\left[\sin \left(x\right)\right]$$