If 5-y^2=x^2 then find d^2y/dx^2 at the point (2, 1) in simplest form.
1 answer:
Answer:
y"(2, 1) = -5
Step-by-step explanation:
Step 1: Define implicit differentiation
5 - y² = x²
Step 2: Find dy/dx
- Take implicit differentiation: -2yy' = 2x
- Isolate y': y' = 2x/-2y
- Isolate y': y' = -x/y
Step 3: Find d²y/dx²
- Quotient Rule: y'' = [y(-1) - y'(-x)] / y²
- Substitute y': y" = [-y - (-x/y)(-x)] / y²
- Simplify: y" = [-y - x²/y] / y²
- Multiply top/bottom by y: y" = (-y² - x²) / y³
- 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
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