Question

Find dy/dx, assuming a, b, c are constants. a) xy + x + y = 5 b) x^2y - 2y + 5 = 0

Answer 1

To find dy/dx for each equation, we use implicit differentiation. For xy + x + y = 5, dy/dx is found to be -(y + 1)/(x + 1). For x^2y - 2y + 5 = 0, dy/dx is -2xy/(x^2 - 2).

Step-by-step explanation:

To find dy/dx, we need to differentiate both sides of the given equations concerning x while assuming that y is a function of x. This is a problem that involves implicit differentiation because y is not isolated on one side of the equation.

Equation (a):

Starting with xy + x + y = 5, we differentiate both sides with respect to x:

1. For the term xy, we apply the product rule: d/dx(xy) = y + x(dy/dx).
2. The derivatives of x and y concerning x are 1 and dy/dx, respectively.
3. Combining these, we get x(dy/dx) + y + 1 + dy/dx = 0.
4. Solving for dy/dx, we find dy/dx = -(y + 1)/(x + 1).

Equation (b):

For x^2y - 2y + 5 = 0, we follow a similar process:

1. For term x^2y, we apply the product rule: d/dx(x^2y) = 2xy + x^2(dy/dx).
2. The derivative of -2y with respect to x is -2(dy/dx).
3. Putting it all together, we get 2xy + x^2(dy/dx) - 2(dy/dx) = 0.
4. Factor out dy/dx: dy/dx(x^2 - 2) + 2xy = 0.
5. Finally, we solve for dy/dx: dy/dx = -2xy/(x^2 - 2).