Find the derivative of the function y defined implicitly in terms of x.
y = √(xy+ 9)
This is the second time you have posted a question that I had already answered.
http://www.jiskha.com/display.cgi?id=1340367899
I had asked for a clarification in the previous post.
Are you even reading our replies??
To find the derivative of a function defined implicitly in terms of another variable, we can use the implicit differentiation method.
Step 1: Begin by differentiating both sides of the equation with respect to x.
d/dx [y] = d/dx [√(xy + 9)]
Step 2: Apply the chain rule to the right-hand side of the equation.
d/dx [y] = (1/2) [(xy + 9)^(-1/2)] * d/dx [xy + 9]
Step 3: Differentiate the right-hand side term by term.
d/dx [xy + 9] = d/dx [xy] + d/dx [9]
= y * d/dx [x] + 0 (since 9 is a constant)
= y * 1 + 0
= y
Step 4: Substitute the result from step 3 into step 2.
d/dx [y] = (1/2) [(xy + 9)^(-1/2)] * y
Step 5: Simplify the expression.
d/dx [y] = (y/2) / √(xy + 9)
Therefore, the derivative of the function y defined implicitly in terms of x is (y/2) / √(xy + 9).
To find the derivative of the function y implicitly in terms of x, we can use the implicit differentiation method.
The first step is to differentiate both sides of the equation with respect to x. This means that we differentiate y with respect to x and also the right-hand side of the equation.
Let's start by differentiating the left-hand side of the equation. Since y is a function of x, we can use the chain rule. The derivative of √u is 1/2√u times the derivative of u, where u = xy + 9. Applying the chain rule, we get:
d/dx (y) = d/dx (√(xy + 9))
= 1/2√(xy + 9) * d/dx (xy + 9)
Now, let's differentiate the right-hand side of the equation. The derivative of xy + 9 with respect to x is simply y + x * dy/dx. Note that we're treating y as a function of x in this case, so we need to use the product rule: differentiating the first term (xy) gives y, and differentiating the second term (9) gives 0.
Now we have:
d/dx (y) = 1/2√(xy + 9) * (y + x * dy/dx)
To find dy/dx, we can solve this equation for dy/dx by moving the rest of the terms to the other side:
dy/dx - x * (1/2√(xy + 9)) * dy/dx = (1/2√(xy + 9)) * y
Now we can factor out dy/dx:
(1 - x * (1/2√(xy + 9))) * dy/dx = (1/2√(xy + 9)) * y
Finally, we can solve for dy/dx by dividing both sides by (1 - x * (1/2√(xy + 9))):
dy/dx = ((1/2√(xy + 9)) * y) / (1 - x * (1/2√(xy + 9)))
So, the derivative of the function y implicitly in terms of x is given by:
dy/dx = ((1/2√(xy + 9)) * y) / (1 - x * (1/2√(xy + 9)))