find \\( \\frac { d y } { d x } \\) by implicit differentiation.\n\\( y = 4 x ^ { 2 } + 9 y x \\)\n\\(…

Explanation:

Step1: Differentiate both sides with respect to (x)

Differentiate (y = 4x^{2}+9yx) term - by - term. Using the sum rule ((u + v)^\prime=u^\prime + v^\prime), where (u = 4x^{2}) and (v = 9yx). The derivative of (y) with respect to (x) is (\frac{dy}{dx}), the derivative of (4x^{2}) using the power rule ((x^{n})^\prime=nx^{n - 1}) is (8x). For the term (9yx), use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 9y) and (v = x). So ((9yx)^\prime=9\frac{dy}{dx}\cdot x+9y\cdot1). We get (\frac{dy}{dx}=8x + 9x\frac{dy}{dx}+9y).

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

Rearrange the terms to isolate (\frac{dy}{dx}) on one side. (\frac{dy}{dx}-9x\frac{dy}{dx}=8x + 9y). Factor out (\frac{dy}{dx}) on the left - hand side: (\frac{dy}{dx}(1 - 9x)=8x + 9y). Then (\frac{dy}{dx}=\frac{8x + 9y}{1 - 9x}).

Answer:

(\frac{8x + 9y}{1 - 9x})