Use implicit differentiation to find the slope of the tangent line to the curve y/(x-2y) = y^3 +9 (1,10/21)
ive done this problem around 10 times and i still cant get it....
when i did it the last time i got -9-y^3/ 3y^2 x - 8y^3 - 19
and the slope around 0.4754 but it is still wrong i don't know what i am doing wrong..
The point (1, 10/21) is not on the curve. Plug in x = 1 and y = 10/21 and you will see that the equation is not satisfied. Are you sure you stated the problem correctly?
Looks like it should be
y/(x-2y) = x^3 +9
To find the slope of the tangent line to the curve at the given point, we need to use implicit differentiation. Let's go through the steps together:
1. Start with the given equation: y / (x - 2y) = y^3 + 9.
2. Differentiate both sides of the equation with respect to x. We treat y as a function of x and use the chain rule for the terms involving y. Remember, when differentiating y with respect to x, we need to include dy/dx.
d/dx(y / (x - 2y)) = d/dx(y^3 + 9).
3. For the left-hand side, we use the quotient rule:
[(x - 2y) * (d/dx(y))] - [y * (d/dx(x - 2y))] / (x - 2y)^2 = 3y^2 * (dy/dx).
4. Simplify the left-hand side first:
[(x - 2y) * (dy/dx)] - [y * (1 - 2(dy/dx))] / (x - 2y)^2 = 3y^2 * (dy/dx).
Now, we need to isolate dy/dx, which represents the slope of the tangent line.
5. Multiply both sides by (x - 2y)^2:
(x - 2y) * [(x - 2y) * (dy/dx)] - y * (1 - 2(dy/dx)) = 3y^2 * (dy/dx) * (x - 2y)^2.
Simplify the left-hand side further:
(x - 2y)^2 * (dy/dx) - (2y - y + 2y * (dy/dx)) = 3y^2 * (dy/dx) * (x - 2y)^2.
Expand and group similar terms:
(x - 2y)^2 * (dy/dx) - (y + 2y * (dy/dx)) = 3y^2 * (dy/dx) * (x - 2y)^2.
Combine like terms:
(x - 2y)^2 * (dy/dx) - 3y * (dy/dx) = 3y^2 * (dy/dx) * (x - 2y)^2.
6. Factor out (dy/dx):
[(x - 2y)^2 - 3y] * (dy/dx) = 3y^2 * (dy/dx) * (x - 2y)^2.
7. Divide both sides by [(x - 2y)^2 - 3y]:
(dy/dx) = [3y^2 * (dy/dx) * (x - 2y)^2] / [(x - 2y)^2 - 3y].
Now we have the expression for dy/dx (the slope of the tangent line) in terms of x and y. To find the slope at the point (1, 10/21), substitute x = 1 and y = 10/21 into the expression for dy/dx.
Let's plug in the values and calculate the slope.