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Use the equation 2 tan⁻¹(x²y) = x xy² to find dy/dx.

User Leiiv
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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.

User Francesco Rigoni
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