Question

find the derivative of y with respect to t.
y = sin^(-1)(sqrt(11)t)
dy/dt = □
(simplify your answer. type an exact answer, using radicals as needed)

Explanation:

Step1: Recall derivative formula

The derivative of $y = \sin^{-1}(u)$ with respect to $t$ is $\frac{dy}{dt}=\frac{1}{\sqrt{1 - u^{2}}}\cdot\frac{du}{dt}$ by the chain - rule. Here $u = \sqrt{11}t$.

Step2: Find $\frac{du}{dt}$

Since $u=\sqrt{11}t$, then $\frac{du}{dt}=\sqrt{11}$ (using the power - rule $\frac{d}{dt}(at)=a$ for a constant $a$).

Step3: Substitute $u$ and $\frac{du}{dt}$ into the chain - rule formula

Substitute $u = \sqrt{11}t$ and $\frac{du}{dt}=\sqrt{11}$ into $\frac{dy}{dt}=\frac{1}{\sqrt{1 - u^{2}}}\cdot\frac{du}{dt}$. We get $\frac{dy}{dt}=\frac{\sqrt{11}}{\sqrt{1-(\sqrt{11}t)^{2}}}$.

Step4: Simplify the denominator

$(\sqrt{11}t)^{2}=11t^{2}$, so $\frac{dy}{dt}=\frac{\sqrt{11}}{\sqrt{1 - 11t^{2}}}$.

Answer:

$\frac{\sqrt{11}}{\sqrt{1 - 11t^{2}}}$