Differentiate the following functions with respect to x: a) y = x b) y = -x2 + 3 c) y = x3/3 + x2/2 + x d) y = 3x7 − 7x3 + 21x2 e) y = x4/4 - x3/3 + x2/2 - x + 3 f) y = sqrt(x5)

$$\frac{dy}{dx} = 1 \cdot x^{1-1} = 1 \cdot x^0 = 1$$

The derivative is $\boxed{\mathbf{1}}$.

b) $y = -x^2 + 3$

$$\frac{dy}{dx} = -2x^{2-1} + 0 = -2x$$

The derivative is $\boxed{\mathbf{-2x}}$.

c) $y = \frac{x^3}{3} + \frac{x^2}{2} + x$

$$\frac{dy}{dx} = x^2 + x + 1$$

The derivative is $\boxed{\mathbf{x^2 + x + 1}}$.

d) $y = 3x^7 - 7x^3 + 21x^2$

$$\frac{dy}{dx} = 21x^6 - 21x^2 + 42x$$

The derivative is $\boxed{\mathbf{21x^6 - 21x^2 + 42x}}$.

e) $y = \frac{x^4}{4} - \frac{x^3}{3} + \frac{x^2}{2} - x + 3$

$$\frac{dy}{dx} = x^3 - x^2 + x - 1$$

The derivative is $\boxed{\mathbf{x^3 - x^2 + x - 1}}$.

f) $y = \sqrt{x^5}$

$$\frac{dy}{dx} = \frac{5}{2}x^{\frac{5}{2}-1} = \frac{5}{2}x^{\frac{3}{2}}$$

The derivative is $\boxed{\mathbf{\frac{5}{2}x^{3/2}}}$.

g) $y = \frac{1}{x^{2/5}}$

$$\frac{dy}{dx} = -\frac{2}{5x^{7/5}}$$

The derivative is $\boxed{\mathbf{-\frac{2}{5x^{7/5}}}}$.

Part (B)

a) $U = (2x + 3)^2$

Let $u = 2x+3$.

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

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