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Let's break down the given problem.

### Provided information:
- \(\frac{d}{dx}(x^2) = 2x\)
- \(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\)
- \[
\frac{d}{dx} \left(9x^2 + 5y^2 - x\right)^2 = 2(9x^2 + 5y^2 - x)\left(18x + 10y \frac{dy}{dx} - 1\right)
\]

### Required:
- Apply the derivatives to the following:
  \[
  \frac{d}{dx} (x^2) + \frac{d}{dx} (y^2) = \frac{d}{dx} \left( (9x^2 + 5y^2 - x)^2 \right)
  \]

### Solution:
1. From the left side:
   \[
   \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = 2x + 2y \frac{dy}{dx}
   \]

2. From the right side:
   \[
   \frac{d}{dx} \left( (9x^2 + 5y^2 - x)^2 \right) = 2(9x^2 + 5y^2 - x)\left(18x + 10y \frac{dy}{dx} - 1\right)
   \]

When you compare the two sides:
- The term next to the blank spot would be \(10y\) as it's derived from \(\frac{d}{dx}(5y^2)\) inside the expression.

Thus, the expression should be:
\[
2x + 2y \frac{dy}{dx} = 2(9x^2 + 5y^2 - x)(18x + 10y \frac{dy}{dx} - 1)
\]

Would you like more detailed explanations or do you have any questions on any steps?

Here are some related questions:
1. How would implicit differentiation work if \(y\) is a function of \(x\)?
2. How does the chain rule apply when differentiating a composition of functions?
3. Can you explain the product rule and how it applies here?
4. How can you simplify expressions after differentiating implicitly?
5. What is the significance of differentiating both sides of an equation in problems involving curves?

**Tip:** When differentiating implicitly, always apply the chain rule whenever a variable like \(y\) is dependent on \(x\).

We have determined the following: \(\frac{d}{dx}(x^2) = 2x\), \(\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}\), and we need to apply the derivatives to find \(\frac{d}{dx}((9x^2 + 5y^2 - x)^2)\).

Learn how to apply implicit differentiation to expressions involving multiple variables, using the chain rule and product rule. This example covers differentiating expressions like \(\frac{d}{dx}((9x^2 + 5y^2 - x)^2)\), commonly found in undergraduate calculus problems.

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