How do you find #(d^2y)/(dx^2)# given #x^2y+xy^2=6#?

2 Answers
Mar 30, 2017

#(d^2y)/(dx^2)=(4y(2x+y)(x+y))/(x^2(x+2y)^2)-(2xy^2(2x+y)^2)/(x(x+2y))^3-(2y)/(x(x+2y)#

Explanation:

Generally differentiation problems involve functions i.e. #y=f(x)# - written explicitly as functions of #x#. However, some functions y are written implicitly as functions of #x#.

So what we do is to treat #y# as #y=y(x)# and use chain rule. This means differentiating #y# w.r.t. #y#, but as we have to derive w.r.t. #x#, as per chain rule, we multiply it by #(dy)/(dx)#.

Here we are given #x^2y+xy^2=6#

hence differentiating it we get

#2x xxy+x^2xx(dy)/(dx)+x xx2y(dy)/(dx)+y^2=0#

or #2xy+x^2(dy)/(dx)+2xy(dy)/(dx)+y^2=0# ................(A)

i.e. #(dy)/(dx)=-(2xy+y^2)/(x^2+2xy)=-(y(2x+y))/(x(x+2y))#

Differentiating (A) further, we have

#2x(dy)/(dx)+2y+x^2(d^2y)/(dx^2)+2x(dy)/(dx)+2y(dy)/(dx)+2x((dy)/(dx))^2+2xy(d^2y)/(dx^2)+2y(dy)/(dx)=0#

or #(d^2y)/(dx^2)(x^2+2xy)+(dy)/(dx)(4x+4y)+2x((dy)/(dx))^2+2y=0#

or #(d^2y)/(dx^2)(x(x+2y))=-4(dy)/(dx)(x+y)-2x((dy)/(dx))^2-2y#

= #4(y(2x+y))/(x(x+2y))(x+y)-2x((y(2x+y))/(x(x+2y)))^2-2y#

and #(d^2y)/(dx^2)=1/(x(x+2y))(4(y(2x+y))/(x(x+2y))(x+y)-2x((y(2x+y))/(x(x+2y)))^2-2y)#

= #(4y(2x+y)(x+y))/(x^2(x+2y)^2)-(2xy^2(2x+y)^2)/(x(x+2y))^3-(2y)/(x(x+2y)#

Mar 31, 2017

# (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 #y# explicitly as a function of #x#, only implicitly through the equation #F(x, y) = 0# which defines #y# as a function of #x, y = y(x)#. Therefore we can write #F(x, y) = 0# as #F(x, y(x)) = 0#. Differentiating both sides of this, using the partial chain rule gives us

# (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 # F(x,y) = x^2y + xy^2 - 6 #; Then;

#(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 # G(x,y) =-(2xy+y^2)/(x^2+2xy) #; Then:

# (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 #