To find the derivative \(\frac{dy}{dx}\) of the equation \((3x - 7x^2)^8 = \frac{1}{y^2}\), we will use implicit differentiation.

Given:
\[
(3x - 7x^2)^8 = \frac{1}{y^2}
\]

### Step 1: Differentiate both sides with respect to \(x\).

#### Differentiating the left side:
\[
\text{Let } u = 3x - 7x^2
\]
Then the left side is \(u^8\), and using the chain rule:
\[
\frac{d}{dx} \left( u^8 \right) = 8u^7 \cdot \frac{du}{dx}
\]
\[
\frac{du}{dx} = \frac{d}{dx} \left( 3x - 7x^2 \right) = 3 - 14x
\]
So, 
\[
\frac{d}{dx} \left( (3x - 7x^2)^8 \right) = 8(3x - 7x^2)^7 \cdot (3 - 14x)
\]

#### Differentiating the right side:
Using the chain rule on \(\frac{1}{y^2}\):
\[
\frac{d}{dx} \left( \frac{1}{y^2} \right) = \frac{d}{dy} \left( \frac{1}{y^2} \right) \cdot \frac{dy}{dx}
\]
\[
\frac{d}{dy} \left( \frac{1}{y^2} \right) = -\frac{2}{y^3}
\]
So,
\[
\frac{d}{dx} \left( \frac{1}{y^2} \right) = -\frac{2}{y^3} \cdot \frac{dy}{dx}
\]

### Step 2: Set the derivatives equal to each other.
\[
8(3x - 7x^2)^7 \cdot (3 - 14x) = -\frac{2}{y^3} \cdot \frac{dy}{dx}
\]

### Step 3: Solve for \(\frac{dy}{dx}\).
\[
\frac{dy}{dx} = -\frac{8y^3(3x - 7x^2)^7 \cdot (3 - 14x)}{2}
\]
\[
\frac{dy}{dx} = -4y^3(3x - 7x^2)^7 \cdot (3 - 14x)
\]

Thus, the derivative \(\frac{dy}{dx}\) is:
\[
\boxed{\frac{dy}{dx} = -4y^3(3x - 7x^2)^7 \cdot (3 - 14x)}
\]

Would you like further details or have any questions?

Here are some related questions:

1. How does the chain rule apply to the differentiation of composite functions?
2. What is implicit differentiation, and when is it used?
3. Can you explain how to find the derivative of an equation involving both \(x\) and \(y\)?
4. How do you handle the differentiation of functions raised to a power?
5. What are the key steps in solving an equation for \(\frac{dy}{dx}\) when both \(x\) and \(y\) are involved?
6. How does implicit differentiation differ from explicit differentiation?
7. Can you explain the process of differentiating \(\frac{1}{y^2}\) with respect to \(x\)?
8. What happens if the equation involves higher powers of \(y\) or more complex expressions?

**Tip:** When dealing with complex implicit differentiation, carefully apply the chain rule and keep track of all terms involving \( \frac{dy}{dx} \) to solve correctly.

Learn how to find the derivative dy/dx of the equation (3x - 7x^2)^8 = 1/y^2 using implicit differentiation. This problem involves advanced calculus concepts including implicit differentiation and the chain rule, suitable for students in Grades 11-12.

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