Question

Find dy/dx by implicit differentiation. x³y -xy³= 4

Answer 1

To find dy/dx for the equation x³y - xy³ = 4, perform implicit differentiation, applying the product rule and the chain rule, then solve for dy/dx.

Step-by-step explanation:

To find dy/dx by implicit differentiation for the equation x³y - xy³ = 4, follow these steps:

1. Differentiate both sides of the equation with respect to x. Remember that y is a function of x, so you need to apply the product rule and the chain rule.
2. For the term x³y, the product rule gives us 3x²y + x³(dy/dx), and for -xy³, it gives us -y³ - 3x²y(dy/dx).
3. After differentiating, collect all terms with dy/dx on one side of the equation and the rest on the other side.
4. Factor out dy/dx and solve for dy/dx.

The differentiated equation should look like this:

3x²y + x³(dy/dx) - y³ - 3x²y(dy/dx) = 0

Finally, solving for dy/dx, you should find dy/dx in terms of x and y.