https://file.aiquickdraw.com/mathbot/file/1726722377181ecstavcm.jpg

The problem asks for the second derivative \( \frac{d^2y}{dx^2} \) given that:

- \( x = t^2 \)
- \( y = t^3 \)

Let's solve it step by step.

### Step 1: Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)

- \( x = t^2 \)
  \[
  \frac{dx}{dt} = 2t
  \]

- \( y = t^3 \)
  \[
  \frac{dy}{dt} = 3t^2
  \]

### Step 2: Find \( \frac{dy}{dx} \) using the chain rule

We know that:
\[
\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3t^2}{2t} = \frac{3t}{2}
\]

### Step 3: Differentiate \( \frac{dy}{dx} \) again to find \( \frac{d^2y}{dx^2} \)

Now we need to differentiate \( \frac{dy}{dx} = \frac{3t}{2} \) with respect to \( x \). Using the chain rule again:
\[
\frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{3t}{2} \right) = \frac{d}{dt} \left( \frac{3t}{2} \right) \cdot \frac{dt}{dx}
\]

First, differentiate \( \frac{3t}{2} \) with respect to \( t \):
\[
\frac{d}{dt} \left( \frac{3t}{2} \right) = \frac{3}{2}
\]

Next, we use the fact that \( \frac{dx}{dt} = 2t \), so \( \frac{dt}{dx} = \frac{1}{2t} \).

Now we can compute:
\[
\frac{d^2y}{dx^2} = \frac{3}{2} \cdot \frac{1}{2t} = \frac{3}{4t}
\]

### Final Answer:
The correct answer is \( \frac{3}{4t} \), which matches option **(B)**.

---

Would you like further details on any part of this solution? 

Here are 5 related questions you could explore:
1. How does the chain rule work in parametric differentiation?
2. What happens if \( x \) and \( y \) are defined in terms of different parameters?
3. Can the second derivative tell us about concavity in parametric equations?
4. How do we find higher-order derivatives in parametric equations?
5. What are the geometric interpretations of first and second derivatives in parametric curves?

**Tip:** Always remember to apply the chain rule when differentiating parametric equations!

If x = t^2 and y = t^3, then d^2y/dx^2 is equal to:

This problem explores parametric differentiation to find the second derivative of y = t^3 with respect to x = t^2. Applying the chain rule step-by-step, the solution guides you through calculating dy/dx and then d^2y/dx^2. Suitable for Grade 12 or early university students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation