How do you find the second derivative by implicit differentiation?

Question

Emma Aldridge · Accepted Answer

Let us find #{d^2y}/{dx^2}# for #x^3+y^3=1#. First, let us find #{dy}/{dx}#.
#x^3+y^3=1#
by differentiating with respect to #x#,
#Rightarrow 3x^2+3y^2{dy}/{dx}=0#
by subtracting #3x^2#,
#Rightarrow3y^2{dy}/{dx}=-3x^2#
by dividing by #3y^2#,
#Rightarrow {dy}/{dx}=-{x^2}/{y^2}# Now, let us find #{d^2y}/{dx^2}#.
by differentiating with respect to #x#,
#Rightarrow{d^2y}/{dx^2}=-{2x cdot y^2-x^2 cdot 2y{dy}/{dx}}/{(y^2)^2}
=-{2x(y^2-xy{dy}/{dx})}/{y^4}#
by plugging in #{dy}/{dx}=-{x^2}/{y^2}#,
#Rightarrow{d^2y}/{dx^2}=-{2x[y^2-xy(-x^2/y^2)]}/y^4=-{2x(y^2+x^3/y)}/y^4#
by multiplying the numerator and the denominator by #y#,
#Rightarrow{d^2y}/{dx^2}=-{2x(y^3+x^3)}/y^5#
by plugging in #y^3+x^3=1#,
#Rightarrow{d^2y}/{dx^2}=-{2x}/y^5#

Evelyn Agard · Answer

undefined To find the second derivative by implicit differentiation, follow these steps:

1. Differentiate the given equation with respect to the variable of interest, treating the other variables as constants.
2. Solve the resulting equation for the first derivative.
3. Differentiate the first derivative obtained in step 2 with respect to the variable of interest to find the second derivative.