To find the derivative of the equation $\sec(y) = x^3$ using implicit differentiation, follow these steps:

1. Differentiate both sides of the equation with respect to $x$, remembering that $y$ is a function of $x$ (so apply the chain rule when differentiating $y$).

[ \frac{d}{dx}[\sec(y)] = \frac{d}{dx}[x^3] ]

2. The derivative of $\sec(y)$ with respect to $x$ involves the chain rule. The derivative of $\sec(y)$ with respect to $y$ is $\sec(y)\tan(y)$, but since $y$ is a function of $x$, we must multiply by $\frac{dy}{dx}$ (the derivative of $y$ with respect to $x$).

$\frac{d}{dx}[\sec(y)] = \sec(y) \tan(y) \cdot \frac{dy}{dx}$

3. The derivative of $x^3$ is straightforward:

$\frac{d}{dx}[x^3] = 3x^2$

4. Now, combine both sides:

$\sec(y) \tan(y) \cdot \frac{dy}{dx} = 3x^2$

5. To isolate $\frac{dy}{dx}$, divide both sides by $\sec(y) \tan(y)$:

$\frac{dy}{dx} = \frac{3x^2}{\sec(y) \tan(y)}$

6. You could also simplify $\sec(y) \tan(y)$ in terms of trigonometric functions, but this is the implicit derivative of $y$ with respect to $x$.

Would you like to see further simplifications or explanations?

Here are 5 related questions to explore:

1. What is implicit differentiation, and how does it differ from regular differentiation?
2. How would you find the second derivative of the same equation?
3. How do we simplify trigonometric derivatives involving secant and tangent?
4. What is the significance of applying the chain rule in implicit differentiation?
5. How can we check if a given function requires implicit differentiation?

Tip: Implicit differentiation is useful whenever you have an equation where $y$ is a function of $x$ but is not isolated.