Understanding and Calculating the Derivative of sin^2(7x)

Understanding and Calculating the Derivative of sin^2(7x)

Introduction

Understanding and calculating derivatives is a critical part of calculus. In this article, we will explore the derivative of the function y sin2(7x). We will use different methods to solve for the derivative and discuss their implications for understanding trigonometric functions and their applications in various fields.

Method 1: The Chain Rule

The first method to find the derivative of sin2(7x) is by using the chain rule. The chain rule is a fundamental technique in differentiation, especially useful when dealing with composite functions. Let's walk through the steps:

Step 1: Identifying the Outer and Inner Functions

In the function y sin2(7x), the outer function is sin2(u) and the inner function is 7x. Here, u 7x.

Step 2: Applying the Chain Rule

The chain rule states that if we have a composite function y f(g(x)), then the derivative is given by:

dy/dx f'(g(x)) * g'(x)

For y sin2(7x):

dy/dx d/dx [sin2(7x)] 2sin(7x) * cos(7x) * 7

Step 3: Simplifying the Expression

Using the double-angle identity for sine, which states that 2sin(x)cos(x) sin(2x), we can simplify the expression:

dy/dx 14sin(7x)cos(7x) 7sin(14x)

Therefore, the derivative of sin^2(7x) using the chain rule is:

dy/dx 14sin(7x)cos(7x) 7sin(14x)

Method 2: Using Trigonometric Identities

An alternative method involves using trigonometric identities to simplify the function before differentiation. Let's explore this approach:

Step 1: Using the Identity for sin2(θ)

The identity sin2(θ) (1 - cos(2θ))/2 can be used to rewrite the function:

y sin2(7x) (1 - cos(14x))/2

Step 2: Differentiating the Simplified Expression

Now, differentiate the expression with respect to x:

dy/dx d/dx[(1 - cos(14x))/2] 1/2 * d/dx [1 - cos(14x)]

Since the derivative of a constant is zero and the derivative of cos(14x) is -14sin(14x):

dy/dx 1/2 * (-14sin(14x)) -7sin(14x)

This simplifies to:

dy/dx 7sin(14x)

Conclusion

Both methods lead to the same conclusion that the derivative of sin2(7x) is 14sin(7x)cos(7x), which simplifies to:

dy/dx 7sin(14x)

This result can be verified through trigonometric identities and the chain rule. Understanding these methods is crucial for solving more complex problems in calculus and for applying calculus in various scientific and engineering fields.

Keywords

derivative, sin^2, 7x, calculus, trigonometric functions

References

1. Stewart, J. (2007). Calculus: Early Transcendentals. Belmont, CA: Brooks/Cole.

2. Larson, R., Edwards, B. (2008). Calculus: An Applied Approach. Boston, MA: Houghton Mifflin Company.

3. Farlow, S. J. (1997). Differential Equations and Linear Algebra. New York: Dover Publications.

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