Question

Find \(\frac{d^{999}}{d x^{999}}(\cos x)\)

Short answer

The 999th derivative of \(\cos x\) is \(\sin x\).

Step-by-step solution

Identify the Cycle

The derivatives for \(\cos x\) follow a predictable cycle every four steps. To find where the 999th derivative falls in this cycle, divide 999 by the length of the cycle, which is 4. This gives 249 cycles, and a leftover of 3.

Match the Remainder to the Cycle

The remainder tells us which term in the cycle the 999th derivative falls on. If the remainder was 1, we'd be on the \(-\sin x\) part of the cycle. If the remainder was 2, we'd be on the \(-\cos x\) part. For a 3, we're on the \(\sin x\) part. If the remainder was 0, it would fall on the \(\cos x\) term.

Identify the 999th Derivative

From the above steps, the remainder 3 corresponds to \(\sin x\), so the 999th derivative is \(\sin x\).

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field constitutes a major part of modern mathematics education. Derivatives are a fundamental concept in calculus, representing the rate at which a function changes at any given point. Taking derivatives helps us understand the behavior of functions and find slopes of tangent lines, velocities, and other rates of change.

In practical scenarios, calculus is applied in diverse fields such as physics, engineering, economics, statistics, and even in medicine. For instance, engineers use calculus to design curves in roads, determine how materials will behave under stress, and calculate the optimal dimensions for various shapes or objects.

In this exercise, we apply principles of calculus to find the 999th derivative of the trigonometric cosine function, a process involving both the understanding of derivatives and the cyclic nature of trigonometric derivatives.

Derivatives Cycle

The concept of a derivatives cycle is particularly pertinent when dealing with periodic functions such as trigonometric functions. Periodic functions repeat their values in regular intervals or cycles, and this inherent property is also reflected in their derivatives.

For the trigonometric functions sine and cosine, their derivatives follow a particular cycle of four steps: For \(\cos x\), the cycle is \(\cos x\) to \(−\sin x\) to \(−\cos x\) to \(\sin x\), and then back to \(\cos x\). Knowing this cycle allows us to predict the n-th derivative without having to compute each derivative step by step.

Understanding and using the derivatives cycle is crucial in simplifying complex calculus problems, and it serves as a powerful tool for those studying the field.

Trigonometric Functions Differentiation

Differentiating trigonometric functions is an essential operation in calculus. These functions, which include sine (\(\sin x\)), cosine (\(\cos x\)), and tangent (\(\tan x\)), among others, have specific rules for their derivatives.

The process of differentiating trigonometric functions is based on their periodic properties and symmetries. The derivatives of the sine and cosine functions are particularly important: The derivative of \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\). These derivatives continue to alternate, reflecting the cyclical nature of trigonometric functions—a key concept in solving problems involving higher order derivatives.

Expanding our understanding beyond the first derivative, we can explore higher order derivatives which are derivatives taken multiple times. As we saw in the original exercise, by recognizing patterns and cycles, differentiation becomes a manageable process.

Cosine Function

The cosine function, denoted as \(\cos x\), is one of the primary trigonometric functions. It represents the x-coordinate of a point on the unit circle as it moves counterclockwise from the angle x, measured in radians, from the positive x-axis. Similar to other trigonometric functions, it is periodic with a period of \(2\pi\) radians, meaning it repeats its values every \(2\pi\) radians.

The derivative of \(\cos x\) is \(-\sin x\), and it itself evolves as we take higher order derivatives, following the pattern of the derivatives cycle. For instance, the second derivative of \(\cos x\) with respect to x is \(-\cos x\), and the third derivative is \(\sin x\), subsequently refreshing the cycle. In the context of the given exercise, this cyclical nature allows us to leap directly to the 999th derivative without extensive computation.