Derivative of sin^nx with Respect to x: A Comprehensive Guide

Understanding the Derivative of sin^n(x)

When tackling the derivative of sin^n(x) with respect to x, one must utilize the chain rule and a careful application of basic derivative rules. This guide provides a step-by-step explanation, including how to derive the formula and its practical application.

Defining the Problem

Let us consider the function sin^n(x), where n is a positive integer. The derivative with respect to x requires a thorough understanding of how the function behaves under differentiation. Specifically, we aim to find d/dx (sin^n(x)).

Applying the Chain Rule

To find the derivative, we will use the chain rule, which is a method for differentiating composite functions. The chain rule states that if you have a function y f(g(x)), the derivative of y with respect to x is given by:

dy/dx df/dg · dg/dx

In our case, we can define u sin(x), and therefore, y u^n. Now, we can apply the chain rule as follows:

Step 1: Differentiate with Respect to u

d du^n/du n u^(n-1)

Step 2: Differentiate u with Respect to x

du/dx is the derivative of sin(x) with respect to x, which we know is cos(x).

Step 3: Combine the Results Using the Chain Rule

Combining these results, we can write:

dy/dx n u^(n-1) · cos(x)

Now, substituting back u sin(x), we get:

dy/dx n (sin(x))^(n-1) · cos(x)

This expression can be written more compactly as:

d/dx (sin^n(x)) n cos(x) sin^(n-1)(x)

Simplified Expression and Verification

A simpler way to write the derivative is:

d/dx (sin^n(x)) n cos(x) (sin(x))^(n-1)

This derivative can be verified by simplifying the expression. For instance, if n 1, the derivative is 0, which is the derivative of sin(x). Similarly, for n 2, the derivative is 2 cos(x) sin(x).

Applications of the Chain Rule

The chain rule, as demonstrated in this example, is a fundamental tool in calculus. It is particularly useful in various mathematical contexts, such as:

Optimization problems Differential equations Physics and engineering, where complex functions of time or space need to be differentiated

Conclusion

Understanding the derivative of sin^n(x) is crucial in various fields, including calculus, physics, and engineering. Utilizing the chain rule, we can effectively find the derivative, which is of the form: