If the tangent to the curve $y = f(x)$ at a point is perpendicular to the line $2x - 3y = 5$,then the value of $\frac{dy}{dx}$ at that point is:

If a line having slope $2$ is a tangent to the curve $y=x^4-6x^3+13x^2-10x+5$ at points $P(x_1, y_1)$ and $Q(x_2, y_2)$, where $x_1, x_2 \in \mathbb{N}$, then $x_1x_2 - y_1y_2 =$

If the tangent to the curve $y=x+\sin y$ at a point $(a, b)$ is parallel to the line joining $\left(0, \frac{3}{2}\right)$ and $\left(\frac{1}{2}, 2\right),$ then

Find the equation of the normal at the point $(a m^{2}, a m^{3})$ for the curve $a y^{2}=x^{3}$.

The equation of the tangent to the curve $y=x^3-2x+7$ at the point $(1,6)$ is

Length of the subtangent at $(x_{1}, y_{1})$ on $x^{n} y^{m}=a^{m+n}, m, n>0$,is