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exercises 3.8 implicit differentiation
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use implicit differentiation to determine \\( \frac{d y}{d x} \\) given the
equation \\( x^{6}+y^{5}=-2 \\).
\\( \frac{d y}{d x}= \\)
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Explanation:

Step1: Differentiate both sides

Differentiate \(x^{6}+y^{5}=-2\) with respect to \(x\).
Using the power rule \(\frac{d}{dx}(x^{n}) = nx^{n - 1}\) and the chain - rule \(\frac{d}{dx}(y^{n})=ny^{n - 1}\frac{dy}{dx}\).
For the left - hand side: \(\frac{d}{dx}(x^{6}+y^{5})=\frac{d}{dx}(x^{6})+\frac{d}{dx}(y^{5})\).
\(\frac{d}{dx}(x^{6}) = 6x^{5}\), and \(\frac{d}{dx}(y^{5})=5y^{4}\frac{dy}{dx}\).
The right - hand side: \(\frac{d}{dx}(-2)=0\).
So, \(6x^{5}+5y^{4}\frac{dy}{dx}=0\).

Step2: Solve for \(\frac{dy}{dx}\)

Subtract \(6x^{5}\) from both sides: \(5y^{4}\frac{dy}{dx}=-6x^{5}\).
Then divide both sides by \(5y^{4}\) (\(y
eq0\)).

Answer:

\(\frac{dy}{dx}=-\frac{6x^{5}}{5y^{4}}\)