23–51. Calculating derivatives Find the derivative of the following functions.
y = cos² x

Verified step by step guidance

1. Step 1: Recognize that the function y = \( \cos^2 x \) can be rewritten using the identity \( y = (\cos x)^2 \).
2. Step 2: Apply the chain rule for differentiation, which states that if you have a composite function \( y = (u(x))^n \), then the derivative \( \frac{dy}{dx} = n(u(x))^{n-1} \cdot \frac{du}{dx} \).
3. Step 3: Identify \( u(x) = \cos x \) and \( n = 2 \) in the function \( y = (\cos x)^2 \).
4. Step 4: Differentiate \( u(x) = \cos x \) to find \( \frac{du}{dx} = -\sin x \).
5. Step 5: Substitute \( u(x) \), \( n \), and \( \frac{du}{dx} \) into the chain rule formula to find the derivative: \( \frac{dy}{dx} = 2(\cos x)^{2-1} \cdot (-\sin x) \).

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when differentiating functions like y = cos²(x), where one function is nested within another.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental functions in mathematics that relate angles to side lengths in right triangles. In calculus, these functions have specific derivatives that are essential for solving problems involving rates of change. For example, the derivative of cos(x) is -sin(x), which is crucial when differentiating functions involving cosine.