use implicit differentiation to find \\( \\frac { d y } { d x } \\).\n\\( x ^ { 5 } + y ^ { 5 } = 25 x y…

Explanation:

Step1: Differentiate both sides

Differentiate (x^{5}+y^{5}=25xy) with respect to (x). Using the power rule ((x^{n})^\prime = nx^{n - 1}) and the chain rule ((y^{n})^\prime=ny^{n - 1}\frac{dy}{dx}) for the left - hand side, and the product rule ((uv)^\prime = u^\prime v+uv^\prime) (where (u = 25x) and (v = y)) for the right - hand side. (\frac{d}{dx}(x^{5})+\frac{d}{dx}(y^{5})=\frac{d}{dx}(25xy)) (5x^{4}+5y^{4}\frac{dy}{dx}=25y + 25x\frac{dy}{dx})

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

Group the terms with (\frac{dy}{dx}) on one side: (5y^{4}\frac{dy}{dx}-25x\frac{dy}{dx}=25y - 5x^{4}) Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(5y^{4}-25x)=25y - 5x^{4}) Then (\frac{dy}{dx}=\frac{25y - 5x^{4}}{5y^{4}-25x}) Simplify the fraction by dividing numerator and denominator by (5): (\frac{dy}{dx}=\frac{5y - x^{4}}{y^{4}-5x})

Answer:

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