Differentiate the following expressions: 8.1.1: 3x - 5x2, 8.1.2: 2/x2 - cuberoot(x7)

For the first term, $\frac{d}{dx}(3x)$:

$$\frac{d}{dx}(3x) = 3 \cdot 1 \cdot x^{1-1} = 3x^0 = 3 \cdot 1 = 3$$

For the second term, $\frac{d}{dx}(5x^2)$:

$$\frac{d}{dx}(5x^2) = 5 \cdot 2 \cdot x^{2-1} = 10x^1 = 10x$$

$$\frac{d}{dx} [3x - 5x^2] = 3 - 10x$$

The derivative is $\boxed{\text{3 - 10x}}$.

8.1.2

To determine $g'(x)$, we first rewrite $g(x)$ using exponent notation, then apply the power rule for differentiation.

Recall that $\frac{1}{x^n} = x^{-n}$ and $\sqrt[m]{x^n} = x^{n/m}$.

$$g(x) = \frac{2}{x^2} - \sqrt[3]{x^7}$$ $$g(x) = 2x^{-2} - x^{7/3}$$

For the first term, $2x^{-2}$:

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

For the second term, $-x^{7/3}$:

$$\frac{d}{dx}(-x^{7/3}) = -\frac{7}{3} \cdot x^{7/3-1} = -\frac{7}{3}x^{7/3-3/3} = -\frac{7}{3}x^{4/3}$$

$$g'(x) = -4x^{-3} - \frac{7}{3}x^{4/3}$$

The derivative is \boxed{\text{-4x^{-3} - \frac{7}{3}x^{4/3}}}.