Differentiate y = log(sin x) with respect to x.

Step-by-step solution

1. 1

   Identify the composition. The function is a logarithm wrapped around a trigonometric function, so it is $\ln u$ with the inner function $u=\sin x$. That calls for the chain rule rather than any log identity.

2. 2

   Write down the chain rule. For $y=\ln u$, $\frac{dy}{dx}=\frac{1}{u}\cdot\frac{du}{dx}.$

3. 3

   Differentiate the inner function. With $u=\sin x$ we get $\dfrac{du}{dx}=\cos x$.

4. 4

   Substitute and simplify. $\frac{dy}{dx}=\frac{1}{\sin x}\cdot\cos x=\frac{\cos x}{\sin x}=\cot x.$

5. 5

   Note the domain and sanity-check. The original function needs $\sin x>0$, so the derivative $\cot x$ is stated on those intervals. At $x=1$ the derivative is $\cot 1\approx 0.642093$, which matches a numerical difference quotient of $\ln(\sin x)$ at $x=1$.