Questions: Assume x and y are functions of t. Evaluate dy/dt for the following. y^2 - 5x^3 = -26; dx/dt = -3, x=2, y=6

Transcript text: Assume x and y are functions of t . Evaluate $\frac{\mathrm{dy}}{\mathrm{dt}}$ for the following. \[ y^{2}-5 x^{3}=-26 ; \frac{d x}{d t}=-3, x=2, y=6 \]

Solution

Solution Steps

Step 1: Differentiate both sides of the equation with respect to $t$

Given $F(y, x) = y^2 - 5*x^3 = -26$, differentiate implicitly to get: $\frac{d}{dt}F(y, x) = \frac{d}{dx}F(y, x) \cdot \frac{dx}{dt} + \frac{d}{dy}F(y, x) \cdot \frac{dy}{dt} = 0$ Which simplifies to: $- 15 x^{2} \cdot \frac{dx}{dt} + 2 y \cdot \frac{dy}{dt} = 0$

Step 2: Solve for $\frac{dy}{dt}$

Solving the above equation for $\frac{dy}{dt}$ gives: $\frac{dy}{dt} = - \frac{45 x^{2}}{2 y}$

Step 3: Substitute known values

Substituting $\frac{dx}{dt} = -3$, $x = 2$, and $y = 6$ into the equation, we get $\frac{dy}{dt} = -15$