Answer:
-dx/dy = (1 + x) √(1 − x²)
Step-by-step explanation:
y = √((1 − x) / (1 + x))
Squaring both sides:
y² = (1 − x) / (1 + x)
Take derivative of both sides (use power rule and chain rule on the left, and quotient rule on the right):
2y dy/dx = [(1 + x)(-1) − (1 − x)(1)] / (1 + x)²
2y dy/dx = (-1 − x − 1 + x) / (1 + x)²
2y dy/dx = -2 / (1 + x)²
y dy/dx = -1 / (1 + x)²
Substitute the expression for y:
√((1 − x) / (1 + x)) dy/dx = -1 / (1 + x)²
Multiply both sides by 1 + x:
√((1 − x) (1 + x)) dy/dx = -1 / (1 + x)
√(1 − x²) dy/dx = -1 / (1 + x)
Solve for dy/dx:
dy/dx = -1 / [ (1 + x) √(1 − x²) ]
The gradient of the normal is the slope of the perpendicular line:
-dx/dy = (1 + x) √(1 − x²)
Here's a graph:
desmos.com/calculator/kbglyjdzaj
