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

Verified step by step guidance

1. Identify the function for which you need to find the derivative: \( y = \sin x + \cos x \).
2. Recall the basic derivatives of trigonometric functions: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).
3. Apply the sum rule for derivatives, which states that the derivative of a sum of functions is the sum of their derivatives.
4. Differentiate each term separately: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).
5. Combine the results to find the derivative of the entire function: \( \frac{dy}{dx} = \cos x - \sin x \).

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 that represents the slope of the tangent line to the curve of the function at any given point. Derivatives are used to find rates of change and can be computed using various rules and techniques.

Trigonometric Functions

Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. In calculus, these functions have specific derivatives: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is essential for differentiating functions involving trigonometric expressions.

Sum Rule of Derivatives

The sum rule of derivatives states that the derivative of a sum of functions is equal to the sum of their derivatives. This means that if you have a function that is the sum of two or more functions, you can differentiate each function separately and then add the results together. This rule simplifies the process of finding derivatives for functions like y = sin x + cos x.