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The problem asks to find \( dy \) when \( y = \sin(7x^2) \).

To differentiate \( y = \sin(7x^2) \), we apply the chain rule.

### Steps:
1. The outer function is \( \sin(u) \), where \( u = 7x^2 \).
2. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \), so we have:
   \[
   \frac{d}{dx}[\sin(7x^2)] = \cos(7x^2) \cdot \frac{d}{dx}[7x^2]
   \]
3. Now, differentiate \( 7x^2 \) with respect to \( x \):
   \[
   \frac{d}{dx}[7x^2] = 14x
   \]
4. Substitute this back into the chain rule:
   \[
   dy = \cos(7x^2) \cdot 14x
   \]

### Final Answer:
\[
dy = 14x \cos(7x^2)
\]

Would you like further clarification or details on any step?

Here are 5 related questions you might explore:
1. How does the chain rule apply to more complex functions like \( y = \sin(f(x)) \)?
2. What is the general form of the chain rule in differentiation?
3. Can you explain the derivative of other trigonometric functions like \( \cos(x) \) or \( \tan(x) \)?
4. How do we differentiate products or compositions of trigonometric and polynomial functions?
5. How can we apply these principles to find higher-order derivatives of similar functions?

**Tip:** Always remember to carefully identify the inner and outer functions when applying the chain rule for differentiation.

Learn how to find the derivative of y = sin(7x^2) using the chain rule. This detailed solution includes differentiation of trigonometric functions and application of the chain rule, suitable for Grades 11-12.

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