Question

X³ - y² = 7, also dx/dt = 0.5. Find dy/dt when y = -1.

Answer 1

To find dy/dt, differentiate the equation x³ - y² = 7 implicitly with respect to t. Substitute the given values and solve for dy/dt. When y = -1, dy/dt = -3.

Step-by-step explanation:

To find dy/dt, we can differentiate the equation x³ - y² = 7 implicitly with respect to t. Since dx/dt = 0.5, we have:

3x²(dx/dt) - 2y(dy/dt) = 0

Substituting the given values y = -1 and dx/dt = 0.5:

3x²(0.5) - 2(-1)(dy/dt) = 0

1.5x² + 2(dy/dt) = 0

2(dy/dt) = -1.5x²

dy/dt = -0.75x²

Now, we need to find the value of x when y = -1. Plugging y = -1 into the original equation:

x³ - (-1)² = 7

x³ - 1 = 7

x³ = 8

x = 2

Substituting x = 2 into dy/dt = -0.75x²:

dy/dt = -0.75(2)²

dy/dt = -0.75(4)

dy/dt = -3

Therefore, when y = -1, dy/dt = -3.