Question

If 5-y^2=x^2 then find d^2y/dx^2 at the point (2, 1) in simplest form. ​

Answer 1

y"(2, 1) = -5

Explanation:

Step 1: Define implicit differentiation

5 - y² = x²

Step 2: Find dy/dx

1. Take implicit differentiation: -2yy' = 2x
2. Isolate y': y' = 2x/-2y
3. Isolate y': y' = -x/y

Step 3: Find d²y/dx²

1. Quotient Rule: y'' = [y(-1) - y'(-x)] / y²
2. Substitute y': y" = [-y - (-x/y)(-x)] / y²
3. Simplify: y" = [-y - x²/y] / y²
4. Multiply top/bottom by y: y" = (-y² - x²) / y³
5. Factor negative: y" = -(y² + x²) / y³

Step 4: Substitute and Evaluate

y"(2, 1) = -(1² + 2²) / 1³

y"(2, 1) = -(1 + 4) / 1

y"(2, 1) = -5/1

y"(2, 1) = -5