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Mar 21, 2024

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Solution by Steps

step 2

The derivative of

\frac{Y^2}{x}

\frac{d}{dx}(\frac{Y^2}{x}) = -\frac{Y^2}{x^2}

, assuming

x \neq 0

and

Y

is constant with respect to

x

step 3

The domain of

x

\{x \in \mathbb{R} : x \neq 0\}

step 4

The range of

Y

given the derivative is \{y \in \mathbb{R} : (y = 0 \text{ and } Y = 0) \text{ or } (y < 0 \text{ and } Y \neq 0)\} ..

step 5

The function

\frac{Y^2}{x}

is even because it is symmetric with respect to the y-axis..

Key Concept

Differentiation using the quotient rule

Explanation

When differentiating a function of the form

\frac{f(x)}{g(x)}

, where

f(x)

and

g(x)

are differentiable functions and

g(x) \neq 0

, the quotient rule is used. The rule states that

\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

. In this case, since

Y

is treated as a constant, its derivative with respect to

x

is zero, simplifying the application of the rule.