differentiate sinx sinx

Guri Bee

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21-10-2024

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Differentiating trigonometric functions can be a challenge for many students, but understanding the rules can help simplify the process. In this article, we’ll delve into the differentiation of the function ( f(x) = \sin(x) \sin(x) ) using the product rule. We'll explore not only how to derive it but also provide additional insights, examples, and resources for further study.

What is the Product Rule?

Before we begin the differentiation process, let's recall the product rule in calculus. The product rule states that if you have two differentiable functions ( u(x) ) and ( v(x) ), the derivative of their product is given by:

[ (u \cdot v)' = u' \cdot v + u \cdot v' ]

Where:

• ( u' ) is the derivative of ( u )
• ( v' ) is the derivative of ( v )

Differentiating ( f(x) = \sin(x) \sin(x) )

Now, let's apply this rule to our function:

1. Identify the Functions:

   • Let ( u = \sin(x) )
   • Let ( v = \sin(x) )

2. Differentiate:

   • The derivative of ( u ) is ( u' = \cos(x) )
   • The derivative of ( v ) is ( v' = \cos(x) )

3. Apply the Product Rule:

   • Using the product rule: [ f'(x) = u' \cdot v + u \cdot v' ] Substitute the derivatives: [ f'(x) = \cos(x) \cdot \sin(x) + \sin(x) \cdot \cos(x) ] This simplifies to: [ f'(x) = 2\cos(x)\sin(x) ]

4. Using a Trigonometric Identity:

   • The expression ( 2\cos(x)\sin(x) ) can be further simplified using the double angle identity: [ 2\sin(x)\cos(x) = \sin(2x) ] Thus, we can express the derivative as: [ f'(x) = \sin(2x) ]

Final Result

The derivative of ( f(x) = \sin(x) \sin(x) ) is:

[ f'(x) = \sin(2x) ]

Practical Examples

To see this derivative in action, consider a few practical examples:

• Example 1: If ( x = \frac{\pi}{4} ) [ f'(\frac{\pi}{4}) = \sin\left(2 \cdot \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{2}\right) = 1 ]

• Example 2: If ( x = 0 ) [ f'(0) = \sin(2 \cdot 0) = \sin(0) = 0 ]

These examples highlight how the derivative behaves at different points, and it can be valuable in scenarios such as motion problems or oscillatory phenomena.

Additional Insights

Understanding the differentiation of ( \sin(x) \sin(x) ) is essential for students studying calculus, especially as it lays the groundwork for more advanced topics like Fourier analysis and harmonic functions.

Further Reading

To deepen your understanding, consider exploring:

• Trigonometric Identities: Understanding how to manipulate trigonometric expressions can greatly enhance your calculus skills.
• Applications of Derivatives: Discover how derivatives are used in real-world scenarios, such as in physics for motion analysis.

Conclusion

Differentiating ( \sin(x) \sin(x) ) demonstrates the effective use of the product rule and the beauty of trigonometric identities in calculus. The final result, ( f'(x) = \sin(2x) ), not only provides the slope of the function at any point ( x ) but also connects to broader mathematical concepts.

By mastering these differentiation techniques, students can build a solid foundation for their future studies in mathematics and related fields.

References

• Calculus textbooks and online resources
• GitHub discussions for various problem-solving strategies

Note: This article builds upon standard mathematical practices and includes insights drawn from general calculus knowledge. For specific Q&A format from GitHub or other sources, individual author attributions would need to be referenced directly.