Find  
d
2
y
d
x
2
  if  
−
9
x
3
−
3
y
3
=
8
 .

We are given the equation:

\[
-9x^3 - 3y^3 = 8
\]

We need to find \(\frac{d^2y}{dx^2}\), the second derivative of \(y\) with respect to \(x\).

### Step 1: Differentiate implicitly
To find the second derivative, we first need to find the first derivative \(\frac{dy}{dx}\). Start by differentiating both sides of the equation with respect to \(x\), using implicit differentiation.

\[
\frac{d}{dx} \left( -9x^3 - 3y^3 \right) = \frac{d}{dx}(8)
\]

On the left-hand side:

- The derivative of \(-9x^3\) is \(-27x^2\).
- The derivative of \(-3y^3\) requires applying the chain rule. The derivative of \(y^3\) with respect to \(y\) is \(3y^2\), and then we multiply by \(\frac{dy}{dx}\), since \(y\) is a function of \(x\).

Thus, we get:

\[
-27x^2 - 9y^2 \cdot \frac{dy}{dx} = 0
\]

### Step 2: Solve for \(\frac{dy}{dx}\)
Rearranging to solve for \(\frac{dy}{dx}\):

\[
-9y^2 \cdot \frac{dy}{dx} = 27x^2
\]

\[
\frac{dy}{dx} = \frac{-27x^2}{-9y^2} = \frac{3x^2}{y^2}
\]

### Step 3: Differentiate again to find the second derivative
Now, differentiate \(\frac{dy}{dx} = \frac{3x^2}{y^2}\) again with respect to \(x\) to find \(\frac{d^2y}{dx^2}\).

Use the quotient rule to differentiate:

\[
\frac{d}{dx} \left( \frac{3x^2}{y^2} \right) = \frac{y^2 \cdot \frac{d}{dx}(3x^2) - 3x^2 \cdot \frac{d}{dx}(y^2)}{(y^2)^2}
\]

The derivative of \(3x^2\) with respect to \(x\) is \(6x\), and the derivative of \(y^2\) with respect to \(x\) is \(2y \cdot \frac{dy}{dx}\) (by the chain rule).

Substituting these into the quotient rule:

\[
\frac{d^2y}{dx^2} = \frac{y^2 \cdot 6x - 3x^2 \cdot 2y \cdot \frac{dy}{dx}}{y^4}
\]

Now substitute \(\frac{dy}{dx} = \frac{3x^2}{y^2}\) into the equation:

\[
\frac{d^2y}{dx^2} = \frac{y^2 \cdot 6x - 3x^2 \cdot 2y \cdot \frac{3x^2}{y^2}}{y^4}
\]

Simplify the terms:

\[
\frac{d^2y}{dx^2} = \frac{6xy^2 - 6x^4 \cdot y}{y^6}
\]

\[
\frac{d^2y}{dx^2} = \frac{6x \left( y^2 - x^4 y \right)}{y^6}
\]

### Final Answer:

\[
\frac{d^2y}{dx^2} = \frac{6x (y^2 - x^4 y)}{y^6}
\]

Would you like a detailed breakdown of any steps or have further questions? Here are some related questions to explore:

1. What is the quotient rule for differentiation?
2. How do you apply implicit differentiation to equations involving both \(x\) and \(y\)?
3. How do you simplify complex fractions in derivatives?
4. Can you find higher-order derivatives using the same method?
5. What is the role of chain rule in implicit differentiation?

**Tip**: When dealing with implicit differentiation, always keep track of derivatives of both \(x\) and \(y\), especially when \(y\) is a function of \(x\).

Learn how to find the second derivative d^2y/dx^2 for the equation -9x^3 - 3y^3 = 8 using implicit differentiation. This problem involves advanced concepts like the quotient rule, chain rule, and implicit differentiation. Suitable for students in Grades 11-12.

Math Problem Statement

Find
d 2 y d x 2 if
− 9 x 3 − 3 y 3

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation