The slope of the tangent to the curve (y - x ^ 5 ) ^ 2 = x (1 + x ^...

$2$\left(y - {x}^{5}\right)\left(\dfrac{dy}{dx} - 5{x}^{4}\right)$<br >$= 1{\left(1 + {x}^{2}\right)}^{2} + \left(x\right)\left(2\left(1 + {x}^{2}\right)\left(2x\right)\right)
Now put $x = 1, y = 3$ and $\dfrac{dy}{dx} = m .$
$2$\left(3 - 1\right)\left(m - 5\right) = 1\left(4\right) + \left(1\right)\left(4\right)\left(2\right)$<br >$m - 5 = \dfrac{12}{4}$<br >$m = 5 + 3 = 8$<br >$\dfrac{dy}{dx} = m = 8 .