Question

Consider the curve x²y² = x² − y³
(a). Find the 1st derivative, dy/dx of the curve.

Answer 1

To find the first derivative dy/dx of the curve x²y² = x² - y³, we apply implicit differentiation, arrange terms involving dy/dx on one side, and then solve for dy/dx.

Step-by-step explanation:

The student has asked to find the first derivative, dy/dx, of the curve x²y² = x² - y³. To do this, we need to use implicit differentiation since the variables x and y are mixed together in the equation.

Starting with the given equation x²y² = x² - y³, we differentiate both sides with respect to x. Remember that when differentiating a y term with respect to x, you multiply by dy/dx to account for the implicit differentiation:

1. First, differentiation of x²y² gives us 2xy² + x²(2y)(dy/dx).
2. Next, differentiation of the right side x² gives us 2x, and for -y³ it's -3y²(dy/dx).
3. Now, equate the two differentiated sides: 2xy² + x²(2y)(dy/dx) = 2x - 3y²(dy/dx).
4. Solve for dy/dx by isolating all the terms involving dy/dx on one side and factoring them out. This will result in dy/dx(2x²y - 3y²) = 2x - 2xy².
5. Finally, we obtain dy/dx by dividing both sides by (2x²y - 3y²).

The result is dy/dx = (2x - 2xy²) / (2x²y - 3y²).

Answer 2

To find the first derivative, dy/dx, of the curve x²y² = x² - y³, we apply implicit differentiation and solve for dy/dx, resulting in dy/dx = (2xy² - 2x) / (2x²y - 3y²).

Step-by-step explanation:

We are going to find the first derivative, dy/dx, of the curve given by the equation x²y² = x² - y³. To do this, we apply implicit differentiation. Differentiating both sides with respect to x, we get:

2xy² + 2x²y(dy/dx) = 2x - 3y²(dy/dx).

Rearranging the terms to solve for dy/dx, we obtain:

dy/dx = (2xy² - 2x) / (2x²y - 3y²).

This gives us the first derivative of the curve in terms of x and y.