Find d²/dx² (sin x + cos x).

Verified step by step guidance

1. First, identify the function you need to differentiate: f(x) = sin(x) + cos(x).
2. Calculate the first derivative, f'(x), by differentiating each term separately. The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Therefore, f'(x) = cos(x) - sin(x).
3. Now, find the second derivative, f''(x), by differentiating f'(x). Differentiate cos(x) to get -sin(x) and differentiate -sin(x) to get -cos(x).
4. Combine the results from the previous step to express the second derivative: f''(x) = -sin(x) - cos(x).
5. Review the process to ensure each differentiation step was applied correctly, confirming that the second derivative of the original function is f''(x) = -sin(x) - cos(x).

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. In this context, we need to differentiate the function sin x + cos x to find its first derivative, which will then be differentiated again to obtain the second derivative.

Second Derivative

The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function and can indicate concavity. In this case, calculating d²/dx² (sin x + cos x) involves taking the derivative of the first derivative to analyze how the rate of change itself is changing.

Trigonometric Derivatives

Trigonometric derivatives are specific rules for differentiating trigonometric functions. For example, the derivative of sin x is cos x, and the derivative of cos x is -sin x. Understanding these derivatives is essential for solving the given problem, as they will be applied to find both the first and second derivatives of the function sin x + cos x.