Find dy/dx y = natural log of sin(x)

Solution:

Step 1:

Apply differentiation to both sides of the equation with respect to $x$: $\frac{d}{dx}(y) = \frac{d}{dx}(\ln(\sin(x)))$.

Step 2:

The derivative of $y$ with respect to $x$ is denoted as $\frac{dy}{dx}$.

Step 3:

Proceed to differentiate the natural logarithm of the sine function.

Step 3.1:

Invoke the chain rule for differentiation: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$, where $f(x) = \ln(x)$ and $g(x) = \sin(x)$.

Step 3.1.1:

Introduce a substitution $u = \sin(x)$ to facilitate the application of the chain rule: $\frac{d}{du}[\ln(u)] \cdot \frac{d}{dx}[\sin(x)]$.

Step 3.1.2:

Compute the derivative of $\ln(u)$ with respect to $u$: $\frac{1}{u} \cdot \frac{d}{dx}[\sin(x)]$.

Step 3.1.3:

Substitute back $u$ with $\sin(x)$: $\frac{1}{\sin(x)} \cdot \frac{d}{dx}[\sin(x)]$.

Step 3.2:

Express $\frac{1}{\sin(x)}$ as $\csc(x)$: $\csc(x) \cdot \frac{d}{dx}[\sin(x)]$.

Step 3.3:

Identify the derivative of $\sin(x)$ with respect to $x$ as $\cos(x)$: $\csc(x) \cdot \cos(x)$.

Step 3.4:

Simplify the expression.

Step 3.4.1:

Rearrange the factors: $\cos(x) \cdot \csc(x)$.

Step 3.4.2:

Rewrite $\csc(x)$ using sine and cosine: $\cos(x) \cdot \frac{1}{\sin(x)}$.

Step 3.4.3:

Combine the cosine and reciprocal sine terms: $\frac{\cos(x)}{\sin(x)}$.

Step 3.4.4:

Translate $\frac{\cos(x)}{\sin(x)}$ into cotangent: $\cot(x)$.

Step 4:

Formulate the final equation by equating the left-hand side to the right-hand side: $y = \cot(x)$.

Step 5:

Replace $y$ with $\frac{dy}{dx}$ to complete the differentiation: $\frac{dy}{dx} = \cot(x)$.

Knowledge Notes:

The problem involves finding the derivative of a function $y$ with respect to $x$, where $y$ is the natural logarithm of the sine of $x$. The solution employs several key concepts in calculus:

1. Differentiation: The process of finding the derivative, which measures how a function changes as its input changes.

2. Chain Rule: A fundamental theorem in calculus used to differentiate compositions of functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

3. Natural Logarithm: The logarithm to the base $e$, where $e$ is Euler's number, approximately equal to 2.71828. The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$.

4. Trigonometric Functions and Their Derivatives: The sine function, $\sin(x)$, has a derivative of $\cos(x)$, and the cosecant function, $\csc(x)$, is the reciprocal of $\sin(x)$. The cotangent function, $\cot(x)$, is the reciprocal of the tangent function and can also be expressed as $\frac{\cos(x)}{\sin(x)}$.

5. Simplification: The process of rewriting an expression in a simpler or more easily usable form. In this case, it involves recognizing and using trigonometric identities to simplify the expression.

By applying these concepts, the problem is solved through a step-by-step differentiation process, leading to the final derivative $\frac{dy}{dx} = \cot(x)$.