If x^3 - y^3 =3 , use only implicit diff. to find d^2y/dx^2 . Express your answer in terms of x and y only and write is as a proper fraction that is completely simplified as much as possible.
I've got dy/dx= x^2/y^2
but couldn't find second derivative.
thanks!
just use the quotient rule:
y" = (2xy^2 - 2x^2y y')/y^4
= 2x(y-xy')/y^3
= 2x(y-x(x^2/y^2))/y^3
= 2x(y^3-x^3)/y^5
or, use implicit differentiation twice:
x^3-y^3 = 3
3x^2 - 3y^2 y' = 0
6x - 6y(y')^2 - 3y^2y" = 0
y^2 y" = 2x - 2y(y')^2
= 2(x -y(x^2/y^2)^2)
= 2(xy^4-x^4y)/y^4
= 2x(y^3-x^3)/y^3
y" = 2x(y^3-x^3)/y^5
To find the second derivative, we first need to differentiate the given equation using implicit differentiation.
The given equation is x^3 - y^3 = 3.
Differentiating both sides with respect to x, we get:
3x^2 - 3y^2 * (dy/dx) = 0.
Now, we can solve this equation for (dy/dx):
dy/dx = (3x^2) / (3y^2).
To find the second derivative, we differentiate the above expression again with respect to x.
Differentiating the expression (dy/dx) = (3x^2) / (3y^2) with respect to x:
(d^2y/dx^2) = [ (d/dx) (3x^2) ] / (3y^2).
Differentiating 3x^2 gives us:
(d/dx) (3x^2) = 6x.
Plugging this back into the original expression, we have:
(d^2y/dx^2) = (6x) / (3y^2).
Finally, we express the answer as a proper fraction that is completely simplified as much as possible:
(d^2y/dx^2) = 2x / y^2.