1. **Problem:** Find the derivative of $y = \sin(3x)$. 2. **Formula:** The derivative of $\sin(u)$ is $\cos(u) \cdot \frac{du}{dx}$. 3. Here, $u = 3x$, so $\frac{du}{dx} = 3$. 4. Therefore, $\frac{dy}{dx} = \cos(3x) \cdot 3 = 3\cos(3x)$. 5. **Problem:** Find the derivative of $y = \cos(5x)$. 6. **Formula:** The derivative of $\cos(u)$ is $-\sin(u) \cdot \frac{du}{dx}$. 7. Here, $u = 5x$, so $\frac{du}{dx} = 5$. 8. Therefore, $\frac{dy}{dx} = -\sin(5x) \cdot 5 = -5\sin(5x)$. 9. **Problem:** Find the derivative of $y = \tan(2x + 1)$. 10. **Formula:** The derivative of $\tan(u)$ is $\sec^2(u) \cdot \frac{du}{dx}$. 11. Here, $u = 2x + 1$, so $\frac{du}{dx} = 2$. 12. Therefore, $\frac{dy}{dx} = \sec^2(2x + 1) \cdot 2 = 2\sec^2(2x + 1)$. 13. **Problem:** Find the derivative of $y = \sec(4x^2)$. 14. **Formula:** The derivative of $\sec(u)$ is $\sec(u)\tan(u) \cdot \frac{du}{dx}$. 15. Here, $u = 4x^2$, so $\frac{du}{dx} = 8x$. 16. Therefore, $\frac{dy}{dx} = \sec(4x^2) \tan(4x^2) \cdot 8x = 8x \sec(4x^2) \tan(4x^2)$. 17. **Problem:** Find the derivative of $y = \sin(x^3)$. 18. **Formula:** The derivative of $\sin(u)$ is $\cos(u) \cdot \frac{du}{dx}$. 19. Here, $u = x^3$, so $\frac{du}{dx} = 3x^2$. 20. Therefore, $\frac{dy}{dx} = \cos(x^3) \cdot 3x^2 = 3x^2 \cos(x^3)$.