Final answer:
Implicit differentiation of (x² y)¶=y involves applying the chain rule, product rule, and power rule. After differentiating both sides with respect to x and applying these rules, one can solve for dy/dx which is expressed in terms of x and y.
Step-by-step explanation:
The student asked about the implicit differentiation of (x² y)¶=y. To differentiate this equation implicitly with respect to x, we must apply the chain rule, product rule, and power rule of differentiation. Here's a step-by-step process:
- First, let's take the derivative of both sides of the equation with respect to x. We are differentiating (x²y)¶ on the left-hand side and y on the right-hand side.
- For the left-hand side, since it is a product within a power, we would need to use the chain rule and the product rule together. The derivative of u¶ (where u = x²y) is 6u⁵ times the derivative of u. We also know that the derivative of x²y with respect to x is 2xy + x²(dy/dx) due to the product rule.
- Now, simplifying and solving for dy/dx, we find that dy/dx is a function of x and y.