Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x

Verified step by step guidance

1. Start by identifying the function y = -2 sin(x). We need to find the fourth derivative of this function with respect to x.
2. Recall that the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These derivatives alternate as you continue differentiating.
3. Calculate the first derivative: y' = d/dx [-2 sin(x)] = -2 cos(x).
4. Calculate the second derivative: y'' = d/dx [-2 cos(x)] = 2 sin(x).
5. Calculate the third derivative: y''' = d/dx [2 sin(x)] = 2 cos(x). Finally, calculate the fourth derivative: y⁽⁴⁾ = d/dx [2 cos(x)] = -2 sin(x).

Key Concepts

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative represents the rate of change of the function, the second derivative indicates the curvature, and so on. In this case, finding the fourth derivative means applying the differentiation process four times to the function y = -2 sin x.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. The sine function, in particular, oscillates between -1 and 1 and has specific derivatives: the derivative of sin x is cos x, and the derivative of cos x is -sin x. Understanding these properties is essential for differentiating functions involving trigonometric terms.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. While not directly applied in this specific question, it is crucial for more complex functions involving trigonometric identities.