91Ó°ÊÓ

Trigonometric functions are a fundamental part of mathematics, often used to describe relationships in right-angled triangles and periodic phenomena. The most common trigonometric functions are the sine, cosine, and tangent functions. For this exercise, we focus primarily on sine and cosine, which are prevalent in calculus problems.

- **Sine Function (\(\sin x\))**: This function gives the y-coordinate of a point on the unit circle corresponding to a given angle \(x\).- **Cosine Function (\(\cos x\))**: This function provides the x-coordinate of that same point on the unit circle.

These functions are periodic with a period of \(2\pi\), meaning they repeat their values every \(2\pi\) units. This periodicity is critical in differentiating them multiple times, as seen in the exercise problem.

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes. In simple terms, it measures how a function's output value changes with respect to changes in the input value. This is crucial in understanding how functions behave, particularly when modeling real-world situations.

To differentiate a function involving trigonometric functions, we must acquaint ourselves with basic derivative rules:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).

In calculus, these rules allow us to calculate the first and subsequent derivatives of trigonometric functions, leading to the solution of complex mathematical problems.

The first derivative of a function is often denoted as \(\frac{dy}{dx}\), and it provides the slope of the tangent line to the function at any given point. For the function \(y = \sin x + \cos x\), finding the first derivative is the initial step in determining the second derivative.

Given \(y = \sin x + \cos x\), applying differentiation rules:- The derivative of \(\sin x\) is \(\cos x\).- The derivative of \(\cos x\) is \(-\sin x\).

Hence, the first derivative \(\frac{dy}{dx} = \cos x - \sin x\). This expression represents the rate at which the original function changes with respect to \(x\), acting as a foundation step to find the second derivative.