Question

If $y = 4\sqrt{x}$, then $\frac{d^2y}{dx^2} =$}

• A. $\frac{8}{y^2} \frac{dy}{dx}$
• B. $\frac{-4}{y^2} \frac{dy}{dx}$
• C. $\frac{-8}{y^2} \frac{dy}{dx}$
• D. $\frac{-2}{y^2} \frac{dy}{dx}$

Hint:

For questions with options relating derivatives, it is often simpler to evaluate the derivative and each option separately rather than trying to transform one expression into another from scratch.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We calculate the first and second derivatives and then try to express the second derivative in terms of the first derivative and $y$.

Step 2: Detailed Explanation:
1. Given $y = 4\sqrt{x} = 4x^{1/2}$.
2. First derivative $\frac{dy}{dx} = 4 \cdot \frac{1}{2} x^{-1/2} = \frac{2}{\sqrt{x}}$.
3. Second derivative $\frac{d^2y}{dx^2} = 2 \cdot (-\frac{1}{2}) x^{-3/2} = -\frac{1}{x\sqrt{x}} = -\frac{1}{x^{3/2}}$.
4. Let's test the given form in Option (C): $\frac{-8}{y^2} \frac{dy}{dx}$.
Sub $y^2 = 16x$ and $\frac{dy}{dx} = \frac{2}{\sqrt{x}}$:
\[ \frac{-8}{16x} \cdot \frac{2}{\sqrt{x}} = \frac{-16}{16x\sqrt{x}} = -\frac{1}{x\sqrt{x}} \]
This matches our calculated $\frac{d^2y}{dx^2}$.

Step 3: Final Answer:
The second derivative is $\frac{-8}{y^2} \frac{dy}{dx}$.