Find the derivative of the implicit function X^3+Y^3+6XY=0
To find the derivative of the implicit function x^3 + y^3 + 6xy = 0, we can use implicit differentiation.
Taking the derivative of each term with respect to x, we have:
d/dx (x^3) + d/dx (y^3) + d/dx (6xy) = d/dx (0)
3x^2 + 3y^2*(dy/dx) + 6x*(dy/dx) + 6y = 0
Rearranging the terms, we have:
3x^2 + 6xy*(dy/dx) + 3y^2*(dy/dx) = -6y
Factoring out the common factor of dy/dx, we have:
dy/dx(6xy + 3y^2) = -3x^2 - 6y
Finally, we can solve for dy/dx:
dy/dx = (-3x^2 - 6y) / (6xy + 3y^2)
To find the derivative of the implicit function X^3 + Y^3 + 6XY = 0, we need to use the implicit differentiation technique. Differentiating both sides of the equation with respect to x, we get:
3X^2 + d/dx(Y^3) + 6Y + 6XdY/dx = 0
Since we are differentiating with respect to x, we treat Y as a function of x. So, d/dx(Y^3) represents the derivative of Y^3 with respect to x, and dY/dx represents the derivative of Y with respect to x.
To find d/dx(Y^3), we apply the chain rule. Let f(u) = u^3, where u = Y(x). The derivative of f(u) with respect to x is given by:
d/dx(f(u)) = d/du(f(u))*du/dx = 3u^2 * dY/dx
Substituting back into the equation, we have:
3X^2 + 3Y^2 * dY/dx + 6Y + 6XdY/dx = 0
Rearranging the equation, we can solve for dY/dx:
dY/dx = (-3X^2 - 6Y) / (3Y^2 + 6X)
So, the derivative of the implicit function X^3 + Y^3 + 6XY = 0 with respect to x is given by:
dY/dx = (-3X^2 - 6Y) / (3Y^2 + 6X)
To find the derivative of the implicit function \(x^3 + y^3 + 6xy = 0\), we can use the partial derivative method.
Step 1: Differentiate both sides of the equation with respect to x. Treat y as a function of x and use the chain rule:
\[3x^2 + \frac{{dy}}{{dx}}(3y^2 + 6x) = 0\]
Step 2: Solve the above equation for \(\frac{{dy}}{{dx}}\) by isolating it on one side.
\[\frac{{dy}}{{dx}} = \frac{{-3x^2}}{{3y^2 + 6x}}\]
So, the derivative of the implicit function \(x^3 + y^3 + 6xy = 0\) with respect to x is \(\frac{{-3x^2}}{{3y^2 + 6x}}\).