Find dy/dt if x=sqrot7/8 and x^2+ y^3=1 and dx/dt=2.
x^2+ y^3=1
2x dx/dt + 3y^2 dy/dt = 0
when x=√(7/8), y = 1/2, so now we have
2√(7/8) * 2 + 1/8 dy/dt = 0
dy/dt = 16√(7/2)
To find dy/dt, we need to differentiate the equation x^2 + y^3 = 1 with respect to t.
Let's start by differentiating both sides of the equation with respect to t:
d/dt(x^2 + y^3) = d/dt(1)
To differentiate x^2 with respect to t, we can use the chain rule:
d/dt(x^2) = d/dx(x^2) * dx/dt
Since dx/dt is given as 2, we can substitute it in:
2x * dx/dt = 0
Now, let's differentiate y^3 with respect to t. Again, we can use the chain rule:
d/dt(y^3) = d/dy(y^3) * dy/dt
We want to find dy/dt, so let's isolate it:
d/dt(y^3) = 3y^2 * dy/dt
Now, substituting the derivatives back into the original equation, we have:
2x * 2 = 3y^2 * dy/dt
Simplifying further, we have:
4x = 3y^2 * dy/dt
To find dy/dt, we can divide both sides by 3y^2:
dy/dt = (4x) / (3y^2)
Now, substitute the value of x and simplify:
dy/dt = (4 * sqrt(7/8)) / (3y^2)
Finally, we need to find the value of y. For that, we can substitute the value of x into the equation x^2 + y^3 = 1:
(sqrt(7/8))^2 + y^3 = 1
7/8 + y^3 = 1
y^3 = 1 - 7/8
y^3 = 1/8
Now, we can substitute the value of y^3 back into dy/dt:
dy/dt = (4 * sqrt(7/8)) / (3 * (1/8))
dy/dt = 4 * 8 * sqrt(7/8) / 3
dy/dt = 32 * sqrt(7/8) / 3
So, dy/dt = 32 * sqrt(7/8) / 3.
To find dy/dt, we can use implicit differentiation on the equation x^2 + y^3 = 1 with respect to time (t) since x, y, and t are all variables.
First, let's differentiate both sides of the equation with respect to t:
d/dt (x^2) + d/dt (y^3) = d/dt (1)
Now, let's differentiate each term separately using the chain rule:
For the first term, d/dt (x^2), we can use the fact that dx/dt = 2:
2x (dx/dt) = 2x (2) = 4x
For the second term, d/dt (y^3), we need to use the chain rule, but first, we need to express y explicitly in terms of x:
From the given equation, x^2 + y^3 = 1, we can solve for y:
y^3 = 1 - x^2
Taking the cube root of both sides:
y = (1 - x^2)^(1/3)
Now, we can differentiate y with respect to x (dy/dx):
dy/dx = d/dx [(1 - x^2)^(1/3)]
To differentiate this expression, we can use the chain rule:
dy/dx = (1/3) * (1 - x^2)^(-2/3) * (-2x)
Finally, we can differentiate dy/dx with respect to t:
d/dt (dy/dx) = d/dt [(1/3) * (1 - x^2)^(-2/3) * (-2x)]
Now, we have dy/dt:
dy/dt = d/dt [(1/3) * (1 - x^2)^(-2/3) * (-2x)]
To find dy/dt, we need the values of x and dx/dt:
Given: x = √(7/8) and dx/dt = 2
Substitute these values into the expression to find dy/dt.