Find the derivative of the following functions.
y = cot x / (1 + csc x)

Verified step by step guidance

1. Step 1: Identify the functions involved. The given function is \( y = \frac{\cot x}{1 + \csc x} \). This is a quotient of two functions: the numerator \( u = \cot x \) and the denominator \( v = 1 + \csc x \).
2. Step 2: Apply the Quotient Rule for derivatives. The Quotient Rule states that if \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \).
3. Step 3: Find the derivative of the numerator \( u = \cot x \). The derivative \( u' = -\csc^2 x \).
4. Step 4: Find the derivative of the denominator \( v = 1 + \csc x \). The derivative \( v' = -\csc x \cot x \).
5. Step 5: Substitute \( u' \), \( v \), \( u \), and \( v' \) into the Quotient Rule formula: \( y' = \frac{(-\csc^2 x)(1 + \csc x) - (\cot x)(-\csc x \cot x)}{(1 + \csc x)^2} \). Simplify the expression to find the derivative.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx and can be calculated using various rules, such as the product rule, quotient rule, and chain rule.

Quotient Rule

The quotient rule is a formula used to find the derivative of a function that is the ratio of two other functions. If you have a function y = u/v, where u and v are both differentiable functions of x, the derivative is given by y' = (v * u' - u * v') / v^2. This rule is essential for differentiating functions like the one in the question, where cot x is divided by (1 + csc x).

Trigonometric Functions and Their Derivatives

Trigonometric functions such as cotangent (cot) and cosecant (csc) have specific derivatives that are crucial for solving problems involving these functions. The derivative of cot x is -csc^2 x, and the derivative of csc x is -csc x * cot x. Understanding these derivatives allows for the effective application of the quotient rule and the overall differentiation process in the given function.