find the derivative of y with respect to t.\n\n( y = arcsin ( sqrt { 13 } t ) )\n\n( \frac { d y } { d t } =…

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = f(g(t))), then (\frac{dy}{dt}=f^{\prime}(g(t))\cdot g^{\prime}(t)). For (y=\arcsin(u)) where (u = \sqrt{13}t), the derivative of (\arcsin(u)) with respect to (u) is (\frac{1}{\sqrt{1 - u^{2}}}), and the derivative of (u=\sqrt{13}t) with respect to (t) is (\sqrt{13}).

Step2: Substitute (u) and simplify

Substitute (u = \sqrt{13}t) into the formula. We get (\frac{dy}{dt}=\frac{\sqrt{13}}{\sqrt{1-(\sqrt{13}t)^{2}}}). Simplify the denominator: (1 - 13t^{2}). So (\frac{dy}{dt}=\frac{\sqrt{13}}{\sqrt{1 - 13t^{2}}}).

Answer:

(\frac{\sqrt{13}}{\sqrt{1 - 13t^{2}}})