Question

find the derivative of y with respect to t.
y = cos^(-1)(√10t)

dy/dt =
(simplify your answer. type an exact answer, using radicals)

Explanation:

Step1: Recall derivative formula

The derivative of $y = \cos^{-1}(u)$ with respect to $u$ is $y^\prime=-\frac{1}{\sqrt{1 - u^{2}}}$, and by the chain - rule, if $y=\cos^{-1}(u)$ and $u = \sqrt{10}t$, then $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$.

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

Given $u=\sqrt{10}t$, then $\frac{du}{dt}=\sqrt{10}$ (since the derivative of $at$ with respect to $t$ is $a$ for a constant $a$).

Step3: Apply the chain - rule

We know $\frac{dy}{du}=-\frac{1}{\sqrt{1 - u^{2}}}$, substituting $u = \sqrt{10}t$ into it, we get $\frac{dy}{du}=-\frac{1}{\sqrt{1-(\sqrt{10}t)^{2}}}$. Then, by the chain - rule $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}=-\frac{1}{\sqrt{1 - 10t^{2}}}\cdot\sqrt{10}=-\frac{\sqrt{10}}{\sqrt{1 - 10t^{2}}}$.

Answer:

$-\frac{\sqrt{10}}{\sqrt{1 - 10t^{2}}}$