differencation of y = root2 sin 2x +pie/4

The given function is:

\[
y = \sqrt{2} \sin(2x) + \frac{\pi}{4}
\]

To differentiate this function with respect to \(x\), we apply standard differentiation rules:

1. **Derivative of \(\sin(2x)\)**: Using the chain rule, the derivative of \(\sin(2x)\) is \(2 \cos(2x)\).
2. **Constant Multipliers**: The constants \(\sqrt{2}\) and \(\frac{\pi}{4}\) are handled separately.

Let's differentiate each term:

\[
\frac{dy}{dx} = \sqrt{2} \cdot \frac{d}{dx}[\sin(2x)] + \frac{d}{dx} \left(\frac{\pi}{4}\right)
\]

The derivative of \(\sin(2x)\) is \(2 \cos(2x)\), and the derivative of the constant \(\frac{\pi}{4}\) is 0.

So, the derivative becomes:

\[
\frac{dy}{dx} = \sqrt{2} \cdot 2 \cos(2x) = 2\sqrt{2} \cos(2x)
\]

Thus, the derivative of \( y = \sqrt{2} \sin(2x) + \frac{\pi}{4} \) is:

\[
\frac{dy}{dx} = 2\sqrt{2} \cos(2x)
\]

Would you like further details or clarification on any step?

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Here are 5 related questions:

1. How do you apply the chain rule in differentiation?
2. What is the derivative of \( \cos(2x) \) with respect to \( x \)?
3. How do you differentiate trigonometric functions with constants?
4. What happens when you differentiate a constant like \( \frac{\pi}{4} \)?
5. Can you explain why the chain rule is necessary for composite functions?

**Tip**: When differentiating trigonometric functions with composite arguments like \( 2x \), always apply the chain rule to account for the inner function!

Explore the step-by-step differentiation of the function y = √2 sin(2x) + π/4 using the chain rule and basic differentiation rules. Learn how to differentiate trigonometric functions and constants, suitable for Grades 11-12 students.

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