Question

If y = x², then dx/dy is

• A. 2x
• B. 2y
• C. $\pm \frac{1}{2 \sqrt{x}}$
• D. $\pm \frac{1}{2 \sqrt{y}}$

Hint:

When differentiating, remember that the reciprocal of \( \frac{dy}{dx} \) gives \( \frac{dx}{dy} \).

Answer 1

The Correct Option is D

Step 1: Understanding the given function.
We are given that \( y = x^2 \). The question asks for \( \frac{dx}{dy} \), the derivative of \( x \) with respect to \( y \).
Step 2: Differentiation.
We begin by differentiating \( y = x^2 \) with respect to \( x \): \[ \frac{dy}{dx} = 2x. \] Now, we need to find \( \frac{dx}{dy} \), which is the reciprocal of \( \frac{dy}{dx} \). Hence, we have: \[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{2x}. \] Since \( y = x^2 \), we can substitute for \( x \) in terms of \( y \): \[ x = \sqrt{y}, \quad \text{so} \quad \frac{dx}{dy} = \pm \frac{1}{2\sqrt{y}}. \] Step 3: Conclusion.
Thus, \( \frac{dx}{dy} = \pm \frac{1}{2\sqrt{y}} \), so the correct answer is (D) \( \pm \frac{1}{2 \sqrt{y}} \).