Find the derivative if x^3 + 4xy^2 - y^4 = 7
x^3 + 4xy^2 - y^4 = 7 --->
d[x^3 + 4xy^2 - y^4 ] = 0 ----->
3x^2 dx + 4y^2 dx + 8yx dy -4 y^3 dy = 0
3x^2 dx + 4y^2 dx =(4y^3 - 8yx) dy -->
3x^2+ 4y^2=(4y^3 - 8yx) dy/dx ---->
dy/dx = (3x^2+ 4y^2)/(4y^3 - 8yx)
To find the derivative of the given equation, we need to apply the implicit differentiation technique.
The given equation is: x^3 + 4xy^2 - y^4 = 7.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(x^3 + 4xy^2 - y^4) = d/dx(7).
3x^2 + 4y^2(dx/dx) + 4x(dy/dx)y^2 - 4y^3(dy/dx) = 0
Simplifying the equation, we get:
3x^2 + 4y^2 + 4xy^2(dy/dx) - 4y^3(dy/dx) = 0.
Step 2: Rearrange the equation to isolate the derivative (dy/dx).
To do this, we collect the terms involving (dy/dx) on one side of the equation and the other terms on the other side:
dy/dx(4xy^2 - 4y^3) = -3x^2 - 4y^2.
Step 3: Divide both sides by (4xy^2 - 4y^3) to solve for dy/dx:
dy/dx = (-3x^2 - 4y^2) / (4xy^2 - 4y^3).
So, the derivative of the equation x^3 + 4xy^2 - y^4 = 7 with respect to x is (-3x^2 - 4y^2) / (4xy^2 - 4y^3).