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To help you with exercise 3 from the image, I'll extract and analyze the text related to it.

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### Exercise 3:
The problem states:

**Determină** \(\frac{dy}{dx}\) știind că y este o funcție implicită a variabilei \(x\), definită de ecuația:
\[ y - 2x - \arcsin\left(\frac{y}{2}\right) = 0. \]

**Translation:**
Determine \(\frac{dy}{dx}\), knowing that \(y\) is an implicit function of the variable \(x\), defined by the equation:
\[ y - 2x - \arcsin\left(\frac{y}{2}\right) = 0. \]

### Solution:
We will use implicit differentiation to solve this.

The equation is:
\[
y - 2x - \arcsin\left(\frac{y}{2}\right) = 0.
\]

1. Differentiate both sides of the equation with respect to \(x\). Remember that \(y\) is a function of \(x\), so we will use the chain rule where necessary.

\[
\frac{d}{dx} \left( y - 2x - \arcsin\left(\frac{y}{2}\right) \right) = \frac{d}{dx}(0).
\]

2. The derivative of \(y\) with respect to \(x\) is \( \frac{dy}{dx} \), the derivative of \(-2x\) is \(-2\), and the derivative of \(\arcsin\left(\frac{y}{2}\right)\) involves the chain rule:

\[
\frac{d}{dx} \arcsin\left(\frac{y}{2}\right) = \frac{1}{\sqrt{1 - \left( \frac{y}{2} \right)^2}} \cdot \frac{1}{2} \cdot \frac{dy}{dx}.
\]

Thus, the derivative of the whole expression is:

\[
\frac{dy}{dx} - 2 - \frac{1}{\sqrt{1 - \left(\frac{y}{2}\right)^2}} \cdot \frac{1}{2} \cdot \frac{dy}{dx} = 0.
\]

3. Simplify the equation to isolate \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} \left( 1 - \frac{1}{2\sqrt{1 - \left( \frac{y}{2} \right)^2}} \right) = 2.
\]

4. Finally, solve for \( \frac{dy}{dx} \):

\[
\frac{dy}{dx} = \frac{2}{1 - \frac{1}{2\sqrt{1 - \left( \frac{y}{2} \right)^2}}}.
\]

This is the derivative \(\frac{dy}{dx}\) for the implicit function.

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Let me know if you'd like more clarification on any step or if you have additional questions! Here are some related questions for further exploration:

1. How do we apply the chain rule in implicit differentiation?
2. What is the derivative of \(\arcsin(x)\)?
3. How would the process change if \(y\) was an explicit function of \(x\)?
4. Can we simplify the result of \( \frac{dy}{dx} \) further?
5. How do we interpret the solution in terms of the geometry of the curve?

**Tip:** Always check if the expression inside functions like arcsine or logarithm stays within the defined domain when solving real-world problems.

Determine \(\frac{dy}{dx}\), knowing that \(y\) is an implicit function of the variable \(x\), defined by the equation: \(y - 2x - \arcsin\left(\frac{y}{2}\right) = 0.\)

Learn how to find the derivative \(\frac{dy}{dx}\) of an implicit function defined by the equation \(y - 2x - \arcsin\left(\frac{y}{2}\right) = 0\). This exercise covers key concepts in calculus, including implicit differentiation, the chain rule, and inverse trigonometric functions. Suitable for students in Grades 11-12.

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