Question

If \( x = at, y = \frac{a}{t} \), then \( \frac{dy}{dx} \) is:

• A. \( t^2 \)
• B. \( -t^2 \)
• C. \( \frac{1}{t^2} \)
• D. \( -\frac{1}{t^2} \)

Hint:

When using the chain rule, always ensure that the derivatives with respect to the independent variable \( t \) are correctly calculated before simplifying.

Answer 1

The Correct Option is D

Step 1: Differentiate \( x \) and \( y \) with respect to \( t \). We are given: \[ x = at, \quad y = \frac{a}{t} \] Differentiate \( x = at \) with respect to \( t \): \[ \frac{dx}{dt} = a \] Differentiate \( y = \frac{a}{t} \) with respect to \( t \): \[ \frac{dy}{dt} = -\frac{a}{t^2} \] Step 2: Apply the chain rule to find \( \frac{dy}{dx} \). Using the chain rule, we get: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \] Substituting the values of \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \): \[ \frac{dy}{dx} = \frac{-\frac{a}{t^2}}{a} \] Step 3: Simplify the expression. \[ \frac{dy}{dx} = -\frac{1}{t^2} \] Final Answer: \[ \boxed{-\frac{1}{t^2}} \]