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A) Given the function:

April 6, 2026Computer Science

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2 parts

-\frac{1}{x^2}

10x + \frac{1}{x^2}

6 steps

Here are the solutions for the differentiation problems:

Given the function

y = \frac{3x+1}{x}

Rewrite the function by splitting the fraction.
$y = \frac{3x}{x} + \frac{1}{x}$ $y = 3 + x^{-1}$
Differentiate each term with respect to $x$ .
Recall that $\frac{d}{dx}(c) = 0$ for a constant $c$ , and $\frac{d}{dx}(x^n) = nx^{n-1}$ .
$\frac{dy}{dx} = \frac{d}{dx}(3) + \frac{d}{dx}(x^{-1})$ $\frac{dy}{dx} = 0 + (-1)x^{-1-1}$ $\frac{dy}{dx} = -x^{-2}$
Express the answer with a positive exponent.
$\frac{dy}{dx} = -\frac{1}{x^2}$
The derivative is $\boxed{-\frac{1}{x^2}}$ .

Given the function

y = 5x^2 - \frac{1}{x}

Rewrite the second term using a negative exponent.
$y = 5x^2 - x^{-1}$
Differentiate each term with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(5x^2) - \frac{d}{dx}(x^{-1})$ $\frac{dy}{dx} = (2)(5)x^{2-1} - (-1)x^{-1-1}$ $\frac{dy}{dx} = 10x - (-1)x^{-2}$ $\frac{dy}{dx} = 10x + x^{-2}$
Express the answer with a positive exponent.
$\frac{dy}{dx} = 10x + \frac{1}{x^2}$
The derivative is $\boxed{10x + \frac{1}{x^2}}$ .

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