Determine y′′ for
x2 + y 4 = 10
Solution: We know that to get the second derivative we required the first derivative and to get that we'll have to do implicit differentiation. Following is the work for that.
2x + 4 y3 y′ = 0
y′ = - x /2 y3
Now, it is the first derivative. We obtain second derivative by differentiating this that will require implicit differentiation again.
y′′ = ( -x ′ /2 y3 )
= - 2 y3 - x (6 y 2 y′) / (2 y3 )2
= -(2 y3 - 6xy 2 y′) /4 y6
= - y - 3xy′ /2 y 4
It is fine in so far as it goes. Though, we would like there to be no derivatives in the answer.
Generally, we don't mind having x's and/or y's in the answer while doing implicit differentiation, however we actually don't like derivatives in the answer. However, we can get rid of the derivative through acknowledging that we know what the first derivative is and substituting it into the second derivative equation. By doing this gives,
y′′ = - ((y - 3xy′) /2 y 3))
= y - 3x ( -x/2y3) /2 y 4
= y + (3 /2)x2 y -3 /2 y 4