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/Users/taoyiwei/Desktop/截屏2024-03-17 00.14.32.png

Mar 17, 2024

/Users/taoyiwei/Desktop/截屏2024-03-17 00.14.32.png

Generated Graph

Solution by Steps

step 2

The quotient rule states that

\frac{d}{dx} \left( \frac{f}{g} \right) = \frac{g \cdot \frac{df}{dx} - f \cdot \frac{dg}{dx}}{g^2}

, where

f = Y^2

and

g = x

step 3

Since

Y

is treated as a constant with respect to

x

\frac{d}{dx} Y^2 = 0

, and

\frac{d}{dx} x = 1

step 4

Applying the quotient rule, we get

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

step 5

Simplifying, we find

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

, provided

x \neq 0

Key Concept

Differentiation using the quotient rule

Explanation

When differentiating a function of the form

\frac{f}{g}

, where

f

and

g

are functions of

x

, the quotient rule is applied to find the derivative.