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To differentiate the given equation \( Ft \) with respect to \( y \), we'll apply the chain rule and product rule where necessary. The equation contains trigonometric functions, which will also require the use of their respective derivatives.

The equation is:
\[
Ft = 2 \left( K_1 \left( \sqrt{L_3^2 + L_4^2 - 2L_3 L_4 \cos \left( 180^\circ - \left(\arcsin \left( \frac{L_5 \sin \theta_5 + y}{L_5} \right) + \theta_1 \right) \right) \right)} - \sqrt{L_3^2 + L_4^2 - 2L_3 L_4 \cos \theta_6} \right) \sin \theta_8 \right) - 2 \left( K_2 \left( \sqrt{L_2^2 + L_6^2 - 2L_2 L_6 \cos \theta_4} - \sqrt{L_2^2 + L_6^2 - 2L_2 L_6 \cos \left( \arcsin \left( \frac{L_5 \sin \theta_5 + y}{L_5} \right) \right)} \right) \sin \theta_2 \right)
\]

### Differentiation Steps:

1. **Identify terms containing \( y \)**: These are within the arcsin functions.
2. **Apply the chain rule**: Differentiate the outer function (arcsin) and then the inner function.
3. **Differentiate arcsin function**:
   \[
   \frac{d}{dy}\left(\arcsin(x)\right) = \frac{1}{\sqrt{1 - x^2}}
   \]
4. **Differentiate inner functions**:
   \[
   \frac{d}{dy}\left(\frac{L_5 \sin \theta_5 + y}{L_5}\right) = \frac{1}{L_5}
   \]
5. **Combine results** using the product and chain rules as necessary.

Let's proceed with the differentiation:

\[
\frac{dFt}{dy} = 2K_1 \left( - \frac{1}{2\sqrt{L_3^2 + L_4^2 - 2L_3 L_4 \cos(\text{inner})}} \right) \left( -2L_3L_4 \sin(\text{inner}) \right) \left( \frac{d}{dy}\left(\text{inner}\right) \right) \sin \theta_8
\]
where the inner part \( \text{inner} \) is \( \arcsin \left( \frac{L_5 \sin \theta_5 + y}{L_5} \right) + \theta_1 \)

\[
- 2K_2 \left( - \frac{1}{2\sqrt{L_2^2 + L_6^2 - 2L_2 L_6 \cos(\text{inner})}} \right) \left( -2L_2L_6 \sin(\text{inner}) \right) \left( \frac{d}{dy}\left(\text{inner}\right) \right) \sin \theta_2
\]

Combining and simplifying these terms gives the derivative of the function with respect to \( y \).

Would you like to see the complete step-by-step differentiation process, or do you have any specific questions about a particular part of this problem?

### Follow-up Questions
1. What specific part of the differentiation process would you like to focus on?
2. Are you interested in exploring the properties of the arcsin function further?
3. Would you like to see an example of a simpler problem involving arcsin differentiation?
4. How familiar are you with the chain rule in differentiation?
5. Would you like to understand how the product rule applies in this context?
6. Are there any particular steps in the differentiation that you found confusing?
7. Would a geometric interpretation of the problem help in understanding the differentiation?
8. Would you like to know about potential applications of such differentiation in real-world scenarios?

**Tip:** When differentiating complex expressions, breaking down the process into smaller, manageable parts can help in understanding and accuracy.

Learn how to differentiate the equation Ft with respect to y using the chain rule, product rule, and trigonometric derivatives. This problem involves complex trigonometric functions and is suitable for advanced undergraduate or graduate students studying calculus.

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