Word Count : 536

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To find the points of inflection of the curve given by the equation: \[ y = \frac{a^2 x}{x^2 + a^2} \] ### Step 1: Compute the first derivative We start by calculating the first derivative \( y' \) of the function using the quotient rule: \[ y' = \frac{(x^2 + a^2) \cdot \frac{d}{dx}(a^2 x) - a^2 x \cdot \frac{d}{dx}(x^2 + a^2)}{(x^2 + a^2)^2} \] First, calculate the derivatives: - \(\frac{d}{dx}(a^2 x) = a^2\) - \(\frac{d}{dx}(x^2 + a^2) = 2x\) Now substitute these into the quotient rule formula: \[ y' = \frac{(x^2 + a^2)(a^2) - a^2 x(2x)}{(x^2 + a^2)^2} \] \[ y' = \frac{a^2(x^2 + a^2) - 2a^2 x^2}{(x^2 + a^2)^2} \] \[ y' = \frac{a^2(x^2 + a^2 - 2x^2)}{(x^2 + a^2)^2} \] \[ y' = \frac{a^2(a^2 - x^2)}{(x^2 + a^2)^2} \] ### Step 2: ___ _________ __________ ___ _________ _______ ____ _______ _______ _____ _______ ______.
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