Final answer:
To find dy/dx from the equation 2 tan⁻¹(x²y) = x xy², apply implicit differentiation using the chain rule for the inverse tangent on the left side and the product rule on the right side, then solve for dy/dx.
Step-by-step explanation:
To find dy/dx using the given equation 2 tan⁻¹(x²y) = x xy², we need to employ implicit differentiation. The differentiation of the left side with respect to x involves using the chain rule together with the derivative of the inverse tangent function, while the differentiation of the right side is a direct application of the product rule and power rule.
Firstly, differentiating both sides with respect to x:
- For the left side, we get 2 d/dx [tan⁻¹(x²y)].
- For the right side, we use the product rule and get x d/dx [xy²] + y² d/dx [x].
After differentiating, we need to solve for dy/dx by separating all the terms that contain dy/dx to one side of the equation, and then factoring it out. This involves algebraic manipulation and potentially the application of trigonometric identities if needed.