To differentiate the equation \(x^2 + 3y^2 = 7\) implicitly, we follow these steps:

### Step 1: Differentiate each term
- The derivative of \(x^2\) with respect to \(x\) is \(2x\).
- The derivative of \(3y^2\) with respect to \(x\) requires the chain rule, as \(y\) is a function of \(x\). Therefore:
  \[
  \frac{d}{dx}(3y^2) = 3 \cdot 2y \cdot \frac{dy}{dx} = 6y \frac{dy}{dx}
  \]
- The derivative of the constant \(7\) is \(0\).

### Step 2: Combine the derivatives
After differentiating, the equation becomes:
\[
2x + 6y \frac{dy}{dx} = 0
\]

### Step 3: Solve for \(\frac{dy}{dx}\)
Rearrange the equation to solve for \(\frac{dy}{dx}\):
\[
6y \frac{dy}{dx} = -2x
\]
\[
\frac{dy}{dx} = \frac{-2x}{6y}
\]
Simplifying:
\[
\frac{dy}{dx} = \frac{-x}{3y}
\]

Thus, the derivative of \(y\) with respect to \(x\) is:
\[
\frac{dy}{dx} = \frac{-x}{3y}
\]

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

Here are five related questions for practice:
1. How would you differentiate \(x^2 + y^2 = 1\) implicitly?
2. What is implicit differentiation, and when is it used?
3. How do you apply the chain rule in implicit differentiation?
4. Can you solve for \(\frac{dy}{dx}\) if the equation was \(x^3 + y^3 = 6\)?
5. How would you find the second derivative, \(\frac{d^2y}{dx^2}\), for the given equation?

**Tip**: Implicit differentiation is essential when dealing with equations where \(y\) is not explicitly written as a function of \(x\).

Learn how to perform implicit differentiation on the equation x^2 + 3y^2 = 7. This step-by-step solution explains how to apply the chain rule and solve for dy/dx. Suitable for students in Grades 11-12 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