How do you find #(d^2y)/(dx^2)# given #x^2y+xy^2=6#?
2 Answers
Explanation:
Generally differentiation problems involve functions i.e.
So what we do is to treat
Here we are given
hence differentiating it we get
or
i.e.
Differentiating (A) further, we have
or
or
=
and
=
# (d^2y)/(dx^2) = {6xy(x^2+xy+y^2)(x+y) }/ (x^2+2xy)^3 #
Explanation:
There is another (often faster) approach using partial derivatives. Suppose we cannot find
# (partial F)/(partial x) (1) + (partial F)/(partial y) dy/dx = 0 => dy/dx = −((partial F)/(partial x)) / ((partial F)/(partial y)) #
And for the second derivative we similarly have:
# dy/dx = G(x,y) => (d^2y)/(dx^2) = (partial G)/(partial x) + (partial F)/(partial y)G(x,y) #
So Let
#(partial F)/(partial x) = 2xy+y^2 \ \ \ # and# \ \ \ (partial F)/(partial y) = x^2+2xy #
And so:
# dy/dx = -(2xy+y^2)/(x^2+2xy) #
So to get the Second derivative;
Let
# (partial G)/(partial x) = -{(x^2+2xy)(2y) - (2xy+y^2)(2x+2y)} / (x^2+2xy)^2 #
# " " = 2y * {2x^2+3xy+y^2 - x^2-2xy} / (x^2+2xy)^2 #
# " " = 2y * {x^2+xy+y^2 } / (x^2+2xy)^2 #
# (partial G)/(partial y) = -{(x^2+2xy)(2x+2y) - (2xy+y^2)(2x)} / (x^2+2xy)^2 #
# " " = 2x*{2xy+y^2-x^2-3xy-2y^2 } / (x^2+2xy)^2 #
# " " = -2x*{xy+x^2+y^2 } / (x^2+2xy)^2 #
And so using the above formula for the second derivative:
# (d^2y)/(dx^2) = (partial G)/(partial x) + (partial F)/(partial y)G(x,y) #
# " " = {2y(x^2+xy+y^2) } / (x^2+2xy)^2 + { 2x(xy+x^2+y^2)(2xy+y^2) } / {(x^2+2xy)^2(x^2+2xy)} #
Which we can simplify to get:
# (d^2y)/(dx^2) = {2y(x^2+xy+y^2)(x^2+2xy) + 2x(xy+x^2+y^2)(2xy+y^2) } / (x^2+2xy)^3 #
# " " = {2xy(x^2+xy+y^2){ (x+2y) + (2x+y) } }/ (x^2+2xy)^3 #
# " " = {6xy(x^2+xy+y^2)(x+y) }/ (x^2+2xy)^3 #