Differenciate implicitly x^3+y^2=27xy
Thank u very much @Anonymous
You are welcome.
I guess you want with respect to x
3 x^2 dx/dx + 2 y dy/dx = 27x dy/dx + 27 y dx/dx
but dx/dx = 1 as you know
so
3 x^2 - 27 y = (27 x - 2 y )dy/dx
so
dy/dx = (3x^2-27y)/(27x-2y)
9y-2x/y^2-27x
To differentiate the equation implicitly, we'll use the chain rule and product rule.
The given equation is: x^3 + y^2 = 27xy.
Step 1: Take the derivative of both sides with respect to x.
d/dx(x^3) + d/dx(y^2) = d/dx(27xy)
Step 2: Apply the chain rule to the terms involving y.
3x^2 + 2yy' = 27xy' + 27y
Step 3: Rearrange the equation to isolate y'.
27xy' - 2yy' = 27y - 3x^2
Step 4: Factor out y' from the left-hand side.
y'(27x - 2y) = 27y - 3x^2
Step 5: Divide both sides by (27x - 2y) to solve for y'.
y' = (27y - 3x^2) / (27x - 2y)
So, the implicit derivative of the equation x^3 + y^2 = 27xy with respect to x is:
y' = (27y - 3x^2) / (27x - 2y)