Find the derivatives of the following functions: 8.2.1 g(x) = −3x4 + 2x; 8.2.2 y = (2x4 + 1)/x2

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Here are the solutions to the problems:

To determine $g'(x)$ , we apply the power rule for differentiation, which states that $\frac{d}{dx}(ax^n) = nax^{n-1}$ .
Given $g(x) = -3x^4 + 2x$ .

Differentiate each term in the expression.
$g'(x) = \frac{d}{dx}(-3x^4) + \frac{d}{dx}(2x)$
Apply the power rule to each term.
For the first term, $\frac{d}{dx}(-3x^4)$ :
$-3 \cdot 4 \cdot x^{4-1} = -12x^3$
For the second term, $\frac{d}{dx}(2x)$ :
$2 \cdot 1 \cdot x^{1-1} = 2x^0 = 2 \cdot 1 = 2$
Combine the results.
$g'(x) = -12x^3 + 2$
The derivative is \boxed{\text{-12x^3 + 2}}.

8.2.2

To determine $\frac{dy}{dx}$ , we first simplify the expression for $y$ by dividing each term in the numerator by the denominator, then apply the power rule for differentiation.
Given $y = \frac{2x^4 + 1}{x^2}$ .
Rewrite $y$ by separating the terms and simplifying exponents.
$y = \frac{2x^4}{x^2} + \frac{1}{x^2}$ $y = 2x^{4-2} + x^{-2}$ $y = 2x^2 + x^{-2}$
Differentiate each term using the power rule $\frac{d}{dx}(ax^n) = nax^{n-1}$ .
For the first term, $2x^2$ :
$\frac{d}{dx}(2x^2) = 2 \cdot 2 \cdot x^{2-1} = 4x^1 = 4x$
For the second term, $x^{-2}$ :
$\frac{d}{dx}(x^{-2}) = -2 \cdot x^{-2-1} = -2x^{-3}$
Combine the results to find $\frac{dy}{dx}$ .
$\frac{dy}{dx} = 4x - 2x^{-3}$
The derivative is \boxed{\text{4x - 2x^{-3}}}.

Find the derivatives of the following functions: 8.2.1 g(x) = -3x4 + 2x; 8.2.2 y = (2x4 + 1)/x2