Question

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

Answer 1

$2\left(y-x^{5}\right)\left(\frac{d y}{d x}-5 x^{4}\right)$
$=1\left(1+x^{2}\right)^{2}+(x)\left(2\left(1+x^{2}\right)(2 x)\right)$
Now put $x=1, y=3$ and $\frac{d y}{d x}=m$.
$2(3-1)(m-5)=1(4)+(1)(4)(2) $
$m-5=\frac{12}{4} $
$m=5+3=8 $
$\frac{d y}{d x}=m=8$