Assume x and y are functions of t. Evaluate dy/dt.
xy - 5x + 2x^3 = -70; dx/dt=-5, x=2, y=-3
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university of new mexico
To evaluate dy/dt, we need to take the derivative of y with respect to t.
Given the equation xy - 5x + 2x^3 = -70, we can rearrange it to isolate y:
xy - 5x + 2x^3 = -70
xy = 5x - 2x^3 - 70
y = (5x - 2x^3 - 70) / x
Now, to find dy/dt, we need to differentiate y with respect to t, using the chain rule. The chain rule states that if y is a function of u, and u is a function of t, then dy/dt = dy/du * du/dt.
In our case, y is a function of x, and x is a function of t. So y is a composition of two functions, and we can use the chain rule as follows:
dy/dt = dy/dx * dx/dt
To find dy/dx, we can differentiate y with respect to x. Applying the quotient rule, we get:
dy/dx = [(1 * (5 - 6x^2)) - (5x - 2x^3 - 70) * 0] / x^2
dy/dx = (5 - 6x^2) / x^2
Now, we need to find dx/dt. It is given that dx/dt = -5.
Finally, we can substitute dy/dx = (5 - 6x^2) / x^2, dx/dt = -5, x = 2, and solve for dy/dt:
dy/dt = (5 - 6*2^2) / 2^2 * -5
dy/dt = (5 - 6*4) / 4 * -5
dy/dt = (5 - 24) / (-20)
dy/dt = -19 / -20
dy/dt = 19/20
Therefore, dy/dt = 19/20.