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The image contains a mathematical problem, written in Hindi and mathematical notation. Let's break down the key elements.

The problem in part (क) states:
\[
\text{यदि } e^{y}(x+1) = 1 \text{ तो दर्शाइये } \frac{d^2 y}{dx^2} = \left(\frac{dy}{dx}\right)^2
\]

This translates to: 
- "If \( e^{y}(x+1) = 1 \), show that \( \frac{d^2 y}{dx^2} = \left( \frac{dy}{dx} \right)^2 \)."

### Solution:

#### Step 1: Start with the given equation
\[
e^y (x + 1) = 1
\]

Solve for \( e^y \):
\[
e^y = \frac{1}{x + 1}
\]

#### Step 2: Take the natural logarithm on both sides
Taking the natural logarithm \( \ln \) on both sides gives:
\[
y = \ln\left( \frac{1}{x + 1} \right)
\]
This simplifies to:
\[
y = -\ln(x + 1)
\]

#### Step 3: First derivative (find \( \frac{dy}{dx} \))
Now, differentiate \( y = -\ln(x + 1) \) with respect to \( x \):
\[
\frac{dy}{dx} = -\frac{1}{x + 1}
\]

#### Step 4: Second derivative (find \( \frac{d^2y}{dx^2} \))
Differentiate \( \frac{dy}{dx} = -\frac{1}{x + 1} \) with respect to \( x \):
\[
\frac{d^2 y}{dx^2} = \frac{1}{(x + 1)^2}
\]

#### Step 5: Show the relation \( \frac{d^2 y}{dx^2} = \left( \frac{dy}{dx} \right)^2 \)
From Step 3, we know that:
\[
\frac{dy}{dx} = -\frac{1}{x + 1}
\]
Now, square both sides:
\[
\left( \frac{dy}{dx} \right)^2 = \left( -\frac{1}{x + 1} \right)^2 = \frac{1}{(x + 1)^2}
\]

Thus, we have shown that:
\[
\frac{d^2 y}{dx^2} = \left( \frac{dy}{dx} \right)^2
\]

Hence, the equation is verified.

---

Would you like further details or clarification on any step?

Here are 5 follow-up questions for deeper understanding:
1. What is the significance of differentiating a logarithmic function?
2. How can you apply this method to other similar problems involving exponentials?
3. What are some common techniques for solving second-order differential equations?
4. Can you explain how chain rule is used when differentiating implicitly?
5. How does logarithmic differentiation simplify problems involving exponents?

**Tip:** When differentiating logarithmic functions, always check if properties of logarithms can simplify your expressions before taking the derivative.

यदि e^y (x + 1) = 1 तो दर्शाइये d^2y/dx^2 = (dy/dx)^2

Explore how to solve a differentiation problem involving exponential and logarithmic functions. Learn how to prove that the second derivative d^2y/dx^2 is equal to (dy/dx)^2 when e^y(x + 1) = 1. This problem involves key concepts of differentiation, logarithms, and exponentials, suitable for Grades 11-12.

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