By computing the first few derivatives and looking for a pattern, find the following derivatives.


a. d⁹⁹⁹/dx⁹⁹⁹ (cos x)

Verified step by step guidance

1. Start by computing the first derivative of cos(x). The derivative of cos(x) with respect to x is -sin(x).
2. Compute the second derivative. The derivative of -sin(x) is -cos(x).
3. Compute the third derivative. The derivative of -cos(x) is sin(x).
4. Compute the fourth derivative. The derivative of sin(x) is cos(x).
5. Notice the pattern: the derivatives cycle every four steps: cos(x), -sin(x), -cos(x), sin(x). Use this pattern to determine the 999th derivative by finding the remainder when 999 is divided by 4.

Key Concepts

Derivatives of Trigonometric Functions

Understanding the derivatives of basic trigonometric functions is essential. The derivative of cos(x) is -sin(x), and the derivative of sin(x) is cos(x). This cyclical pattern continues, with the derivative of -sin(x) being -cos(x), and the derivative of -cos(x) being sin(x). Recognizing this cycle helps in computing higher-order derivatives.

Higher-Order Derivatives

Higher-order derivatives involve taking the derivative of a function multiple times. For trigonometric functions like cos(x), the derivatives repeat in a cycle every four derivatives. This means that the nth derivative can be determined by finding the remainder of n divided by 4, which indicates the position in the cycle.

Pattern Recognition in Derivatives

Identifying patterns in derivatives is crucial for efficiently computing higher-order derivatives. By observing the cyclical nature of the derivatives of cos(x), one can predict the 999th derivative by recognizing that it corresponds to the third position in the cycle, which is -sin(x). This pattern recognition simplifies the computation process significantly.