Derivatives and Differential Equations of Sine Functions

Introduction

Understanding the derivatives and differential equations of sine functions is crucial for students and professionals in mathematics, physics, engineering, and other scientific disciplines. This article delves into the derivatives and differential equations involving sine functions, specifically focusing on functions like Y sin2x and y x sin 2x.

The Derivative of Y sin2x

Let us begin by finding the derivative of the function Y sin2x.

Step-by-Step Derivative Calculation

We can use the chain rule for differentiation. Let u sin x, then Y u2.

Applying the chain rule:

Y' d(Y)/dx d(u2)/du * d(u)/dx

Y' 2u * cos x

Y' 2 sin x * cos x

Using the double angle identity, 2 sin x cos x simplifies to sin(2x):

Y' sin(2x)

Thus, the derivative of Y sin2x is:

Y' sin(2x).

Differentiating y x sin 2x

Now, let's differentiate the function y x sin 2x.

Step-by-Step Derivative Calculation

We will use the product rule for this differentiation:

y' d/dx (x sin 2x) x * d/dx(sin 2x) sin 2x * d/dx(x)

Derivative of sin 2x using the chain rule:

d/dx (sin 2x) cos 2x * d/dx (2x) 2 cos 2x

Simplifying with the product rule:

y' x * 2 cos 2x sin 2x * 1

y' 2x cos 2x sin 2x

Second Derivative of y

To find the second derivative of y, we differentiate y' 2x cos 2x sin 2x.

Step-by-Step Second Derivative Calculation

For the term 2x cos 2x:

d/dx (2x cos 2x) 2 cos 2x 2x * d/dx (cos 2x)

d/dx (cos 2x) -sin 2x * d/dx (2x) -2 sin 2x

d/dx (2x cos 2x) 2 cos 2x - 4x sin 2x

For the term sin 2x:

d/dx (sin 2x) cos 2x * d/dx (2x) 2 cos 2x

Combining these results:

y'' 2 cos 2x - 4x sin 2x 2 cos 2x

y'' 4 cos 2x - 4x sin 2x

Derivative of sin 2x2

Next, let's find the derivative of sin 2x2 using the chain rule.

Step-by-Step Derivative Calculation

Let u 2x2 and v sin u, then:

d/dx (sin 2x2) d/dx (v) d/dx (sin u) * d/dx (u)

d/dx (sin u) cos u

d/dx (u) d/dx (2x2) 4x

d/dx (sin 2x2) cos 2x2 * 4x

The Derivative of Y sin2x/cos2x

Finally, let's find the derivative of Y sin2x/cos2x.

Step-by-Step Derivative Calculation

Let Y tan2x, thus:

d/dx (tan 2x) d/dx (sin 2x / cos 2x)

d/dx (sin 2x / cos 2x) sec22x * d/dx (2x)

d/dx (2x) 2

dY/dx 2 sec22x

Conclusion

In conclusion, understanding the derivatives and differential equations of sine functions is essential for various applications in science and engineering. The chain rule, product rule, and double angle identities are key tools for solving such problems. The functions and derivatives discussed in this article provide a solid foundation for further studies and practical applications.