Assume that x and y are functions of t and x^2+2xy=5. If dx/dt=-3 when x=-1, find dy/dt
x^2+2xy=5
differentiate with respect to t
2x dx/dt + 2x dy/dt + 2y dx/dt = 0
x dx/dt + x dy/dt + y dx/dt = 0
when x = -1 in the original:
1 + 2(-1)y = 5
-2y = 4
y = -2
so we have dx/dt = -3, x = -1, y = -2
then in
x dx/dt + x dy/dt + y dx/dt = 0
-1(-3) + (-1)dy/dt + (-2)(-3) = 0
3 - dy/dt + 6 = 0
dy/dt = 9
check my arithmetic
Well, if we start with x^2 + 2xy = 5, we can differentiate it with respect to t using the chain rule.
So, differentiating both sides, we get:
2x(dx/dt) + 2y(dx/dt) + 2x(dy/dt) = 0.
Plugging in dx/dt = -3 and x = -1, we have:
2(-1)(-3) + 2y(-3) + 2(-1)(dy/dt) = 0.
Simplifying that expression, we get:
6 - 6y - 2(dy/dt) = 0.
Now, let's solve for dy/dt:
2(dy/dt) = 6 - 6y.
dy/dt = (6 - 6y) / 2.
Simplifying that further, we find:
dy/dt = 3 - 3y.
So, the rate of change of y with respect to t is 3 - 3y.
Now that we have an answer, let me throw in a joke for good measure:
Why did the scarecrow win an award?
Because he was outstanding in his field!
To find dy/dt, we need to differentiate both sides of the equation x^2 + 2xy = 5 with respect to t using the product rule.
Differentiating x^2 + 2xy = 5 with respect to t, we get:
2x(dx/dt) + 2y(dx/dt) + 2x(dy/dt) = 0
Since we know dx/dt = -3 when x = -1, we can substitute these values into the equation:
2(-1)(-3) + 2y(-3) + 2(-1)(dy/dt) = 0
6 - 6y - 2(dy/dt) = 0
Rearranging the equation, we have:
2(dy/dt) = 6 - 6y
Dividing both sides by 2, we get:
dy/dt = 3 - 3y/2
So, when dx/dt = -3 and x = -1, dy/dt = 3 - (3*(-1))/2 = 3 + 3/2 = 3 + 3/2 = 9/2.
To find dy/dt, we need to differentiate the given equation with respect to t and then solve for dy/dt.
Let's differentiate x^2 + 2xy = 5 with respect to t, using the chain rule:
d/dt(x^2) + d/dt(2xy) = d/dt(5)
2x(dx/dt) + 2y(dx/dt) + 2x(dy/dt) = 0
Now, substitute the given values dx/dt = -3 and x = -1:
2(-1)(-3) + 2y(-3) + 2(-1)(dy/dt) = 0
6 - 6y - 2(dy/dt) = 0
Rearranging the equation, we get:
-2(dy/dt) = 6y - 6
Dividing both sides by -2, we get:
dy/dt = (6 - 6y) / 2
Therefore, dy/dt = 3 - 3y
So, dy/dt = 3 - 3y.