The derivative of $y = \cos\left(3x^2 + \frac{6}{3} - 5x^3\right)$ can be computed using the chain rule.

Step-by-step solution:

1. Rewrite the expression for simplification: $y = \cos\left(3x^2 + 2 - 5x^3\right)$

2. Let $u = 3x^2 + 2 - 5x^3$. Now, $y = \cos(u)$.

3. Differentiate $y = \cos(u)$ with respect to $u$: $\frac{dy}{du} = -\sin(u)$

4. Now, differentiate $u = 3x^2 + 2 - 5x^3$ with respect to $x$: $\frac{du}{dx} = 6x - 15x^2$

5. Finally, apply the chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\sin\left(3x^2 + 2 - 5x^3\right) \cdot (6x - 15x^2)$

Thus, the derivative of $y$ is: $\frac{dy}{dx} = -\sin\left(3x^2 + 2 - 5x^3\right) \cdot (6x - 15x^2)$

Would you like more details or have any questions?

Related questions:

1. How do you apply the chain rule in calculus?
2. What are some examples of differentiating trigonometric functions?
3. How do you find the second derivative of trigonometric functions?
4. What is the derivative of higher-degree polynomials?
5. Can the derivative of composite functions always be found using the chain rule?

Tip:

When differentiating composite functions, always remember to differentiate the outer function first and then multiply by the derivative of the inner function.