Solve the following differentiation problems using the chain rule.

Here are the solutions using the chain rule:

Basic Polynomial Composition:

1. To find $\frac{dy}{dx}$ for $y = (3x + 2)^5$:

   $$\frac{dy}{dx} = 5(3x + 2)^{5-1} \cdot \frac{d}{dx}(3x + 2)$$ $$\frac{dy}{dx} = 5(3x + 2)^4 \cdot 3$$ $$\boxed{15(3x + 2)^4}$$

2. To differentiate $f(x) = (2x^2 - 5x + 1)^4$:

   $$f'(x) = 4(2x^2 - 5x + 1)^{4-1} \cdot \frac{d}{dx}(2x^2 - 5x + 1)$$ $$f'(x) = 4(2x^2 - 5x + 1)^3 \cdot (4x - 5)$$ $$\boxed{4(4x - 5)(2x^2 - 5x + 1)^3}$$

3. To find $f'(x)$ if $f(x) = (4 - x^2)^3$:

   $$f'(x) = 3(4 - x^2)^{3-1} \cdot \frac{d}{dx}(4 - x^2)$$ $$f'(x) = 3(4 - x^2)^2 \cdot (-2x)$$ $$\boxed{-6x(4 - x^2)^2}$$

Square Root and Fractional Powers:

4. To differentiate $y = \sqrt{5x^2 + 3x - 1} = (5x^2 + 3x - 1)^{1/2}$:

   $$\frac{dy}{dx} = \frac{1}{2}(5x^2 + 3x - 1)^{\frac{1}{2}-1} \cdot \frac{d}{dx}(5x^2 + 3x - 1)$$ $$\frac{dy}{dx} = \frac{1}{2}(5x^2 + 3x - 1)^{-1/2} \cdot (10x + 3)$$ $$\boxed{\frac{10x + 3}{2\sqrt{5x^2 + 3x - 1}}}$$

5. To find $\frac{dy}{dx}$ if $y = (x^3 + 2x)^{1/3}$:

   $$\frac{dy}{dx} = \frac{1}{3}(x^3 + 2x)^{\frac{1}{3}-1} \cdot \frac{d}{dx}(x^3 + 2x)$$ $$\frac{dy}{dx} = \frac{1}{3}(x^3 + 2x)^{-2/3} \cdot (3x^2 + 2)$$ $$\boxed{\frac{3x^2 + 2}{3(x^3 + 2x)^{2/3}}}$$

6. To find the derivative of $f(x) = (1 - 2x)^{-2}$:

   $$f'(x) = -2(1 - 2x)^{-2-1} \cdot \frac{d}{dx}(1 - 2x)$$ $$f'(x) = -2(1 - 2x)^{-3} \cdot (-2)$$ $$f'(x) = 4(1 - 2x)^{-3}$$ $$\boxed{\frac{4}{(1 - 2x)^3}}$$

Trigonometric Functions:

7. To differentiate $y = \sin(4x^2 + 3x)$:

   $$\frac{dy}{dx} = \cos(4x^2 + 3x) \cdot \frac{d}{dx}(4x^2 + 3x)$$ $$\frac{dy}{dx} = \cos(4x^2 + 3x) \cdot (8x + 3)$$ $$\boxed{(8x + 3)\cos(4x^2 + 3x)}$$

8. To find $f'(x)$ if $f(x) = \cos^3(2x) = (\cos(2x))^3$:

   $$f'(x) = 3(\cos(2x))^{3-1} \cdot \frac{d}{dx}(\cos(2x))$$ $$f'(x) = 3\cos^2(2x) \cdot (-\sin(2x)) \cdot \frac{d}{dx}(2x)$$ $$f'(x) = 3\cos^2(2x) \cdot (-\sin(2x)) \cdot 2$$ $$\boxed{-6\cos^2(2x)\sin(2x)}$$

9. To find $\frac{dy}{dx}$ if $y = \tan(x^2 - 5x)$:

   $$\frac{dy}{dx} = \sec^2(x^2 - 5x) \cdot \frac{d}{dx}(x^2 - 5x)$$ $$\frac{dy}{dx} = \sec^2(x^2 - 5x) \cdot (2x - 5)$$ $$\boxed{(2x - 5)\sec^2(x^2 - 5x)}$$

10. To differentiate $y = \sin^2(3x + \pi/4) = (\sin(3x + \pi/4))^2$:

    $$\frac{dy}{dx} = 2\sin(3x + \pi/4) \cdot \frac{d}{dx}(\sin(3x + \pi/4))$$ $$\frac{dy}{dx} = 2\sin(3x + \pi/4) \cdot \cos(3x + \pi/4) \cdot \frac{d}{dx}(3x + \pi/4)$$ $$\frac{dy}{dx} = 2\sin(3x + \pi/4) \cdot \cos(3x + \pi/4) \cdot 3$$ $$\boxed{6\sin(3x + \pi/4)\cos(3x + \pi/4)}$$

Exponential Functions:

11. To find $\frac{dy}{dx}$ if $y = e^{3x^2 + 2x}$: $$\frac{dy}{dx} = e^{3x^2 + 2x} \cdot \frac{d}{dx}(3x^2 + 2x)$$ $$\frac{dy}{dx} = e^{3x^2 + 2x} \cdot (6x + 2)$$ $$$$