Find the derivative of the function y defined implicitly in terms of x.
y = sqrt(xy + 9)
Please help! Thanks! :)
Did you mean "with respect to x" ?
If so then I would square both sides
y^2 = xy + 9
2y dy/dx = x dy/dx + y
2y dy/dx - x dy/dx = y
dy/dx(2y-x) = y
dy/dx = y/(2y-x) such that √(xy-9) ≥ 0
if you want that expression to be in terms of x only, (like you typed)
we have a mess ahead
we would have to solve
y^2 - xy - 9 = 0 for y in terms of x
y = (x ± √(x^2 +36) )/2
and sub that into dy/dx to get only x's
so I will wait for your clarification before I start that messy substitution.
To find the derivative of the function y defined implicitly in terms of x, we can use the chain rule. The chain rule states that if y is a function of u, and u is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x.
In this case, we are given the equation y = sqrt(xy + 9), which is defined implicitly in terms of x. To find the derivative dy/dx, we need to find both dy/du and du/dx.
First, let's find dy/du. To do this, we need to differentiate both sides of the equation with respect to u. Since y is a function of both x and u, we need to use the chain rule.
Differentiating y = sqrt(xy + 9) with respect to u, we have:
dy/du = (1/2) * (xy + 9)^(-1/2) * (xdy/du + y).
Now, let's find du/dx. To do this, we need to differentiate both sides of the equation x = x with respect to x. Since x does not depend on any other variable, the derivative du/dx is simply 1.
Finally, we can use the chain rule to find dy/dx by multiplying dy/du and du/dx:
dy/dx = (dy/du) * (du/dx).
Substituting the respective values we found earlier:
dy/dx = [(1/2) * (xy + 9)^(-1/2) * (xdy/du + y)] * 1.
Simplifying this expression yields the final result for the derivative of y with respect to x, which is:
dy/dx = (xy + 9)^(-1/2) + (1/2) * (xy + 9)^(-1/2) * y.
So, the derivative of the function y, defined implicitly in terms of x, is (xy + 9)^(-1/2) + (1/2) * (xy + 9)^(-1/2) * y.