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1 year ago in Calculus , Differential Geometry By Prajwal Sharma

How to compute dy/dx for y = -x/(a² - x²)?

While reviewing some classical mechanics problems, I encountered this derivative form. I've applied the quotient rule, but my sign simplification keeps tripping me up, leading to a different result than the text's answer. I need a clear, conversational walk-through to see where my algebraic manipulation is going astray, ensuring I grasp the nuance.

 

QuotientRule AlgebraicManipulation

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By Pritam Kapoor Answered 3 months ago

This is a classic sign trap. Let's walk through it calmly. The quotient rule says: derivative = [ (deriv of top)(bottom) - (top)(deriv of bottom) ] / (bottom)². Here, the top is -x, so its derivative is -1. The bottom is (a² - x²), so its derivative is -2x. Plugging in: [ (-1)(a² - x²) - (-x)(-2x) ] / (a² - x²)². The key is simplifying the numerator: -a² + x² - (2x²) becomes -a² - x². So, dy/dx = -(a² + x²) / (a² - x²)².

 

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