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Apr 8, 2024

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Solution by Steps

step 1

To differentiate the function

\frac{Y^2}{x}

with respect to

x

, we apply the quotient rule

step 2

The quotient rule states that

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

, where

f = Y^2

and

g = x

. Since

Y

is treated as a constant with respect to

x

f' = 0

step 3

Applying the quotient rule, we get

\frac{d}{dx}\left(\frac{Y^2}{x}\right) = \frac{0 \cdot x - Y^2 \cdot 1}{x^2}

step 4

Simplifying the expression, we obtain

-\frac{Y^2}{x^2}

, assuming

x \neq 0

Answer

-\frac{Y^2}{x^2}

Key Concept

Differentiation using the quotient rule

Explanation

The quotient rule is used to differentiate ratios of functions, and in this case, it simplifies to

-\frac{Y^2}{x^2}

because the derivative of

Y^2

with respect to

x

is zero.