Problem 39 - Precalculus

Question

Make a conjecture about the derivative by calculating the first few derivatives and observing the resulting pattern. $$\frac{d^{87}}{d x^{87}}[\sin x]$$

Step-by-Step Solution

Answer

The 87th derivative of \( \sin x \) is \( -\sin x \).

Step 1: Calculate the First Derivative

The first step is to differentiate the function \( \sin x \). The derivative of \( \sin x \) is \( \cos x \).

Step 2: Calculate the Second Derivative

Next, differentiate \( \cos x \). The derivative of \( \cos x \) is \( -\sin x \).

Step 3: Calculate the Third Derivative

Now, differentiate \( -\sin x \). The derivative of \( -\sin x \) is \( -\cos x \).

Step 4: Calculate the Fourth Derivative

Differentiate \( -\cos x \). The derivative of \( -\cos x \) returns us back to \( \sin x \).

Step 5: Identify the Pattern

Observe that the derivatives follow the cycle: \( \sin x, \cos x, -\sin x, -\cos x \). This cycle repeats every four derivatives.

Step 6: Calculate the 87th Derivative

Since the pattern repeats every four derivatives, divide 87 by 4 to determine the equivalent position in the cycle. The remainder when 87 is divided by 4 is 3. Therefore, \( \frac{d^{87}}{d x^{87}}[\sin x] = -\sin x \).

Key Concepts

Differentiation

Differentiation is a fundamental process in calculus that involves calculating the derivative of a function. The derivative measures how a function changes as its input changes. In simpler terms, it tells us the rate at which one quantity changes with respect to another. For example, in physics, the derivative of the position function with respect to time is the velocity - it shows how fast the position changes over time.

To perform differentiation, one needs to follow specific rules and apply them to the function in question. In our example, we begin with differentiating the trigonometric function \( \sin x \), where its derivative is the function \( \cos x \). This involves knowing the derivatives of basic functions, which can be found using differentiation rules such as the power rule, product rule, quotient rule, and chain rule.

As you repeatedly differentiate trigonometric functions like \( \sin x \), you'll notice a cyclic repetition in the derivatives, which is an essential aspect involving trigonometric functions.

Trigonometric Functions

Trigonometric functions such as \( \sin x \), \( \cos x \), and \( \tan x \) are vital in mathematics and are frequently used in various sciences. These functions describe the relationships between the angles and lengths in right triangles and have periodic properties that repeat over fixed intervals.

When we differentiate trigonometric functions, it's important to remember the derivatives of these basic functions:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

Knowing these derivatives is foundational for understanding how these functions behave under differentiation. In the process of differentiating \( \sin x \) multiple times, we see these derivatives emerge, leading to cyclic patterns that we can rely on to make predictions about higher-order derivatives.

Cyclic Patterns

Cyclic patterns appear prominently when differentiating trigonometric functions due to their periodic nature. In the example given, after calculating a few derivatives of \( \sin x \), you notice they follow a repeating cycle: \( \sin x \), \( \cos x \), \( -\sin x \), \( -\cos x \). These four derivatives form a pattern that repeats every four derivatives.

Understanding this repetition is extremely useful because it allows us to predict the outcome of higher-order derivatives without performing each step manually. By dividing the order of the derivative (e.g., 87th derivative in this case) by the length of the pattern cycle (which is 4), the remainder tells us the position within the cycle (e.g., a remainder of 3 points to \( -\sin x \)).

This approach not only simplifies the computation but also highlights the beauty and utility of patterns in mathematics, reducing complex calculations to simple, predictable results.