https://file.aiquickdraw.com/mathbot/file/1729904709658gw0sm80p.jpg

To solve the equation:

\[
y (x^2 + 81) = 162
\]

we first find an explicit expression for \( y \), and then calculate \( \frac{dy}{dx} \).

### Step 1: Solve for \( y \)
Divide both sides of the equation by \( x^2 + 81 \):

\[
y = \frac{162}{x^2 + 81}
\]

### Step 2: Calculate \( \frac{dy}{dx} \)
We need to differentiate \( y \) with respect to \( x \):

Let \( y = \frac{162}{x^2 + 81} \).

Use the quotient rule for differentiation, which states:

\[
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2}
\]

Here, \( u = 162 \) (a constant), so \( u' = 0 \), and \( v = x^2 + 81 \), so \( v' = 2x \).

Thus,

\[
\frac{dy}{dx} = \frac{0 \cdot (x^2 + 81) - 162 \cdot 2x}{(x^2 + 81)^2} = \frac{-324x}{(x^2 + 81)^2}
\]

The derivative \( \frac{dy}{dx} \) is therefore:

\[
\frac{dy}{dx} = \frac{-324x}{(x^2 + 81)^2}
\]

Would you like further details or have any other questions?

### Related Questions
1. How do you apply the quotient rule in calculus?
2. What happens to the derivative as \( x \) approaches infinity?
3. How does the derivative behave near \( x = 0 \)?
4. Can you solve the differential equation for different values of \( x \)?
5. What is the significance of finding \( \frac{dy}{dx} \) in practical applications?

### Tip
When differentiating fractions, remember to carefully apply the quotient rule or consider rewriting the function to simplify the differentiation process.

Solve the equation y(x^2 + 81) = 162 for y to find an explicit expression for y and then calculate dy/dx.

This solution provides a step-by-step approach to solving the equation y(x^2 + 81) = 162 for y and finding the derivative dy/dx. Learn how to apply the quotient rule in calculus for differentiation, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation