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.