6. use implicit differentiation to find \\( \\frac{d y}{d x} \\) for \\( y^{5}+y^{3}=5 x^{2} y^{2}+7 \\).
6. use implicit differentiation to find \\( \\frac{d y}{d x} \\) for \\( y^{5}+y^{3}=5 x^{2} y^{2}+7 \\).
Answer
Explanation:
Step1: Differentiate both sides
Differentiate (y^{5}+y^{3}=5x^{2}y^{2}+7) with respect to (x). Using the chain - rule ((u^n)^\prime=nu^{n - 1}u^\prime) for the left - hand side and the product - rule ((uv)^\prime = u^\prime v+uv^\prime) ((u = 5x^{2}), (v=y^{2})) for the right - hand side. The derivative of (y^{5}) with respect to (x) is (5y^{4}\frac{dy}{dx}), the derivative of (y^{3}) with respect to (x) is (3y^{2}\frac{dy}{dx}), the derivative of (5x^{2}y^{2}) with respect to (x) is (5(2xy^{2}+x^{2}\cdot2y\frac{dy}{dx})), and the derivative of (7) with respect to (x) is (0). So we have (5y^{4}\frac{dy}{dx}+3y^{2}\frac{dy}{dx}=10xy^{2}+10x^{2}y\frac{dy}{dx}).
Step2: Solve for (\frac{dy}{dx})
Group the terms with (\frac{dy}{dx}) on one side: (5y^{4}\frac{dy}{dx}+3y^{2}\frac{dy}{dx}-10x^{2}y\frac{dy}{dx}=10xy^{2}). Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(5y^{4}+3y^{2}-10x^{2}y)=10xy^{2}). Then (\frac{dy}{dx}=\frac{10xy^{2}}{5y^{4}+3y^{2}-10x^{2}y}).
Answer:
(\frac{dy}{dx}=\frac{10xy^{2}}{5y^{4}+3y^{2}-10x^{2}y})