Question

Find the second derivative at the point (2,2), given the function below. 2x^4=4y^3

Answer 1

The second derivative at the point (2,2) is 34/9

Explanation:

2x⁴ = 4y³

2x⁴ - 4y³ = 0

We first need to find dy/dx and then d²y/dx²

On differentiating the equation in terms of x

dy/dx = d(2x⁴ - 4y³) / dx

We get,

dy/dx = 2x³/3y²

On differentiating dy/dx we get,

d²y/dx² = 2x²/y² + 8x⁶/9y⁵

$\frac{d^2y}{dx^2} = \frac{2 X 2^2}{2^2} + \frac{8 X 2^6}{9 X 2^5}\\ \\\frac{d^2y}{dx^2} = 2 + \frac{16}{9}$

d²y/dx² = 34/9

Therefore, the second derivative at the point (2,2) is 34/9