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The given equation is:

\[
\left( \frac{dy}{dx} \right)^2 = -2iAe^{i(\omega t)} \cdot \sin(kx)
\]

To differentiate this with respect to \( x \), let's go step by step.

### Step 1: Start with the equation

\[
\left( \frac{dy}{dx} \right)^2 = -2iAe^{i(\omega t)} \cdot \sin(kx)
\]

We need to take the derivative of both sides with respect to \( x \).

### Step 2: Differentiate both sides

First, apply the chain rule to the left side:

\[
\frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^2 \right) = 2 \cdot \frac{dy}{dx} \cdot \frac{d^2y}{dx^2}
\]

Now, differentiate the right-hand side. Only the sine term depends on \( x \), so:

\[
\frac{d}{dx} \left( -2iAe^{i(\omega t)} \sin(kx) \right) = -2iAe^{i(\omega t)} \cdot k \cdot \cos(kx)
\]

### Step 3: Combine the result

Equating both sides:

\[
2 \cdot \frac{dy}{dx} \cdot \frac{d^2y}{dx^2} = -2iAe^{i(\omega t)} \cdot k \cdot \cos(kx)
\]

### Step 4: Solve for \(\frac{d^2y}{dx^2}\)

To solve for the second derivative, \(\frac{d^2y}{dx^2}\), divide both sides by \( 2 \cdot \frac{dy}{dx} \):

\[
\frac{d^2y}{dx^2} = \frac{-2iAe^{i(\omega t)} \cdot k \cdot \cos(kx)}{2 \cdot \frac{dy}{dx}}
\]

Thus, the second derivative in terms of \( x \) is:

\[
\frac{d^2y}{dx^2} = \frac{-iAe^{i(\omega t)} \cdot k \cdot \cos(kx)}{\frac{dy}{dx}}
\]

This is the result after differentiating the given equation.

Would you like further details on any part of the process? Here are some related questions:

1. How does the chain rule apply in differentiation?
2. Can you explain why the sine term changes when differentiating?
3. How does the presence of complex numbers affect differentiation?
4. What are some common applications of second-order derivatives?
5. How does the function \( e^{i(\omega t)} \) behave in differential equations?

**Tip**: Always apply the chain rule carefully when dealing with powers of derivatives!

Find the differentiation of (dy/dx)^2 = -2iAe^(iωt) * sin(kx)

Step-by-step differentiation of the equation (dy/dx)^2 = -2iAe^(iωt) * sin(kx) using the chain rule and the properties of trigonometric and complex exponential functions. Suitable for undergraduate math students.

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