https://file.aiquickdraw.com/mathbot/file/1735110385094r8r4n5wl.jpg

Let us solve the problem step by step. Here's the assertion and reason in the question:

---

**Assertion (A):** If \( y = \sin^{-1}(\sqrt{x}) \), then  
\[ \frac{dy}{dx} = \frac{3\sqrt{x}}{2\sqrt{1-x^3}}. \]

**Reason (R):**  
The derivative of \( \sin^{-1}(x) \) is  
\[ \frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1-x^2}}, \, |x| \leq 1. \]

We need to check whether both (A) and (R) are correct and if (R) is the correct explanation of (A).

---

### Step 1: Verify Assertion (A)

Given \( y = \sin^{-1}(\sqrt{x}) \), we calculate \( \frac{dy}{dx} \).

1. Differentiate \( y = \sin^{-1}(\sqrt{x}) \) using the chain rule:
   \[
   \frac{dy}{dx} = \frac{1}{\sqrt{1-(\sqrt{x})^2}} \cdot \frac{d}{dx}(\sqrt{x}).
   \]

2. Simplify \( 1-(\sqrt{x})^2 \):
   \[
   \frac{dy}{dx} = \frac{1}{\sqrt{1-x}} \cdot \frac{1}{2\sqrt{x}}.
   \]

3. Combine the terms:
   \[
   \frac{dy}{dx} = \frac{1}{2\sqrt{x(1-x)}}.
   \]

**This is different from the derivative given in Assertion (A)** (\( \frac{3\sqrt{x}}{2\sqrt{1-x^3}} \)). Thus, Assertion (A) is **false**.

---

### Step 2: Verify Reason (R)

The formula given for the derivative of \( \sin^{-1}(x) \):
\[
\frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1-x^2}}, \, |x| \leq 1
\]
is **correct**. This is the standard derivative of the inverse sine function.

---

### Step 3: Evaluate the Options

- **Assertion (A) is false** because the derivative calculation provided is incorrect.
- **Reason (R) is true** as it correctly provides the derivative formula for \( \sin^{-1}(x) \).

Thus, the correct answer is:
\[
\boxed{\text{(d) Assertion (A) is false, but Reason (R) is true.}}
\]

---

Would you like a detailed breakdown of the derivative, or do you have further questions? Here are five follow-up questions for you:

1. What is the role of the chain rule in finding derivatives of composite functions?
2. How do we derive the formula for \( \sin^{-1}(x) \) differentiation step by step?
3. Why does the domain of \( \sin^{-1}(x) \) limit \( x \) to \( |x| \leq 1 \)?
4. Can Assertion (A) ever hold true for a different function of \( y \)?
5. What are the other inverse trigonometric derivatives you need to know?

**Tip:** Always verify derivatives step by step to avoid subtle errors in complex expressions!

If y = sin^(-1)(√x), then dy/dx = 3√x / 2√(1 - x^3)?

Learn how to solve the differentiation problem where y = sin^(-1)(√x). This problem involves verifying if dy/dx = 3√x / 2√(1 - x^3) using chain rule and properties of inverse trigonometric derivatives. Suitable for Grades 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation