Question

use implicit differentiation to determine \\(\frac{dy}{dx}\\) for the equation \\(\frac{5}{x} - \frac{3}{y} = 5\\).

Explanation:

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<reasoning>

Rewrite the equation with negative exponents

\[
5x^{-1} - 3y^{-1} = 5
\]

Differentiate both sides with respect to x

\[
-5x^{-2} - 3(-1)y^{-2}\frac{dy}{dx} = 0
\]
\[
-\frac{5}{x^2} + \frac{3}{y^2}\frac{dy}{dx} = 0
\]

Solve for dy/dx

\[
\frac{3}{y^2}\frac{dy}{dx} = \frac{5}{x^2}
\]
\[
\frac{dy}{dx} = \frac{5y^2}{3x^2}
\]
</reasoning>

<answer>
Use implicit differentiation to determine \(\frac{dy}{dx}\) for the equation \(\frac{5}{x} - \frac{3}{y} = 5\).

\(\frac{dy}{dx} =\) <blank>\(\frac{5y^2}{3x^2}\)</blank>
</answer>

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"Mathematics",
"Calculus",
"Implicit Differentiation"
]
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</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Rewrite the equation with negative exponents

\[
5x^{-1} - 3y^{-1} = 5
\]

Differentiate both sides with respect to x

\[
-5x^{-2} - 3(-1)y^{-2}\frac{dy}{dx} = 0
\]
\[
-\frac{5}{x^2} + \frac{3}{y^2}\frac{dy}{dx} = 0
\]

Solve for dy/dx

\[
\frac{3}{y^2}\frac{dy}{dx} = \frac{5}{x^2}
\]
\[
\frac{dy}{dx} = \frac{5y^2}{3x^2}
\]
</reasoning>

<answer>
Use implicit differentiation to determine \(\frac{dy}{dx}\) for the equation \(\frac{5}{x} - \frac{3}{y} = 5\).

\(\frac{dy}{dx} =\) <blank>\(\frac{5y^2}{3x^2}\)</blank>
</answer>

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"Mathematics",
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