Let xy = 5 and { dy }/{ dt } = 3.
Find { dx }/{ dt } when x = 1.
To find { dx }/{ dt }, we can use implicit differentiation.
Given: xy = 5 and { dy }/{ dt } = 3
We need to find { dx }/{ dt } when x = 1.
Start by taking the derivative of both sides of the equation xy = 5 with respect to t:
d/dt (xy) = d/dt (5)
Using the product rule, we can differentiate xy with respect to t:
x(dy/dt) + y(dx/dt) = 0
Since { dy }/{ dt } = 3, we can substitute it into the equation:
x(3) + y(dx/dt) = 0
Substitute the given value of xy = 5:
1(3) + (5)(dx/dt) = 0
3 + 5(dx/dt) = 0
Simplifying the equation:
5(dx/dt) = -3
Now, divide both sides by 5:
(dx/dt) = -3/5
Therefore, { dx }/{ dt } when x = 1 is -3/5.
To find { dx }/{ dt }, we can use implicit differentiation because the equation xy = 5 is not directly solved for y.
Let's start by differentiating both sides of the equation with respect to t, using the chain rule on the left side:
d/dt (xy) = d/dt (5)
Using the product rule on the left side, we get:
x * (dy/dt) + y * (dx/dt) = 0
Since we know that { dy }/{ dt } = 3, we can substitute it into the equation:
x * 3 + y * (dx/dt) = 0
Now we need to find y when x = 1. We can solve the original equation xy = 5 for y:
y = 5/x
Substituting x = 1, we get y = 5/1 = 5.
Plugging the values of x = 1 and y = 5 into the differentiated equation:
1 * 3 + 5 * (dx/dt) = 0
3 + 5 * (dx/dt) = 0
To find { dx }/{ dt }, we can isolate the term (dx/dt) by moving the 3 to the other side:
5 * (dx/dt) = -3
Finally, dividing both sides by 5 gives us the solution:
(dx/dt) = -3/5
Therefore, { dx }/{ dt } is equal to -3/5 when x = 1.
xy = 5 , use the product rule and differentiate with respect to t
x(dy/dt) + y(dx/dt) = 0
when x= 1 ---> (1)y = 5, so y=5
now sub in all that stuff in
x(dy/dt) + y(dx/dt) = 0
(1)(dy/dt) + 5(3) = 0
dy/dt = -15