Question

Find the second derivative at the point (1,2), given the function below. y+3y^3-2x^2=10x+14

Answer 1

Explanation:

`y + 3 {y}^(3) - 2 {x}^(2) = 10x + 14 \\ \therefore \: y + 3 {y}^(3) = 2 {x}^(2) + 10x + 14\\ differentiatig \: w.r.t. \: x \: on \: both \: sides : \\ \\ (dy)/(dx) + 3 * 3 {y}^(2) (dy)/(dx) = 2 * 2x + 10 + 0 \\ \\ \therefore \: (dy)/(dx) + 9{y}^(2) (dy)/(dx) = 4x + 10 \\ \\ differentiatig \: again \: w.r.t. \: x \: on \: \\ both \: sides : \\ (d)/(dx)((dy)/(dx)) + 9 * 2{y} (d)/(dx)((dy)/(dx)) \: = 4 + 0 \\ \\ \therefore \:(d^(2) y)/(dx^(2)) + 18{y} (d^(2)y)/(dx^(2))= 4 \\ \\ \therefore \:(1 + 18{y} )(d^(2)y)/(dx^(2))= 4 \\ \\ \therefore \:(d^(2)y)/(dx^(2)) = \frac{4}{1 + 18{y} } \\ \\ \therefore \:[(d^(2)y)/(dx^(2))] _((1, \: \: 2))= (4)/(1 + 18 * 2) \\ \\ \therefore \:[(d^(2)y)/(dx^(2))] _((1, \: \: 2))= (4)/(37)`