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2 years ago

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

Mathematics

1 answer:

ratelena [41]2 years ago

3 0

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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