The slope of the tangent to the curve $(y - x^{5})^{2} = x (1 + x^{2})^{2}$ at the point $(1, 3)$ is ___

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Solution

$(y - x^{5})^{2} = x (1 + x^{2})^{2}$
$2 (y - x^{5}) (\frac{d y}{d x} - 5 x^{4}) = (1 + x^{2})^{2} + 2 x (1 + x^{2}) .2 x$
At point $(1, 3)$
$2 (3 - 1) (\frac{d y}{d x} - 5) = (1 + 1)^{2} + 2 (1 + 1) .2 \Rightarrow \frac{d y}{d x} = 8$

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