Question

If $y = 4t$ and $x = \dfrac{4}{t}$, then the value of $\dfrac{dy}{dx}$ will be:

• A. $-t^{2}$
• B. $-\dfrac{1}{t^{2}}$
• C. $-\dfrac{1}{t}$
• D. $t^{3}$

Hint:

When $y$ and $x$ are given in terms of a parameter $t$, use $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.

Answer 1

The Correct Option is A

Step 1: Differentiate $y$ with respect to $t$.
\[ y = 4t $\Rightarrow$ \frac{dy}{dt} = 4 \]

Step 2: Differentiate $x$ with respect to $t$.
\[ x = \frac{4}{t} $\Rightarrow$ \frac{dx}{dt} = -\frac{4}{t^{2}} \]

Step 3: Apply the chain rule.
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{4}{-\frac{4}{t^{2}}} = -t^{2} \]

Step 4: Conclusion.
The correct answer is (A) $-t^{2}$.