Question

use implicit differentiation to determine \\( \frac { d y } { d x } \\) given the equation \\( x ^ { 6 } + y ^ { 5 } = 7 \\).

\\( \frac { d y } { d x } = \\)

Explanation:

Step1: Differentiate both sides with respect to \(x\)

Differentiate \(x^{6}+y^{5}=7\) term - by - term.
Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\) and the chain rule \(\frac{d}{dx}(y^{n})=ny^{n - 1}\frac{dy}{dx}\).
We get \(\frac{d}{dx}(x^{6})+\frac{d}{dx}(y^{5})=\frac{d}{dx}(7)\).
\(6x^{5}+5y^{4}\frac{dy}{dx}=0\).

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

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

Answer:

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