If x=2^^-1t and y=2^^-1t (|t|1), then dydx is equal to.

Question

If $x=2^{\csc^{-1}t}$ and $y=2^{\sec^{-1}t}$ $(|t|\ge1)$, then $\dfrac{dy}{dx}$ is equal to.

Accepted Answer

$\dfrac{dx}{dt}=2^{\csc^{-1}t}\ln2\cdot\dfrac{-1}{|t|\sqrt{t^2-1}}$, $\dfrac{dy}{dt}=2^{\sec^{-1}t}\ln2\cdot\dfrac{1}{|t|\sqrt{t^2-1}}$.

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=-\frac{2^{\sec^{-1}t}}{2^{\csc^{-1}t}}=-\frac yx$$

Solutions · 1