Find the derivative of the following functions.
y = cos x/sin x + 1

Verified step by step guidance

1. Step 1: Recognize that the function y = \frac{\cos x}{\sin x} + 1 can be rewritten as y = \cot x + 1, where \cot x is the cotangent function.
2. Step 2: Recall the derivative of the cotangent function: \frac{d}{dx}(\cot x) = -\csc^2 x.
3. Step 3: Differentiate the function y = \cot x + 1 with respect to x. The derivative of a constant (1 in this case) is 0.
4. Step 4: Apply the derivative rule from Step 2 to find the derivative of \cot x, which is -\csc^2 x.
5. Step 5: Combine the results from Steps 3 and 4 to express the derivative of the entire function: \frac{dy}{dx} = -\csc^2 x.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental in calculus, especially when dealing with periodic phenomena. These functions relate angles to ratios of sides in right triangles and are essential for modeling oscillatory behavior. Understanding their derivatives, such as the fact that the derivative of cos(x) is -sin(x) and the derivative of sin(x) is cos(x), is crucial for solving problems involving these functions.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If y = u/v, where u and v are both differentiable functions, the derivative is given by y' = (v * u' - u * v') / v^2. This rule is particularly useful when differentiating functions like y = cos(x)/sin(x) + 1, as it allows for the systematic calculation of the derivative of the quotient of trigonometric functions.