Question

1. find the derivative.

a. ( y=csc ^{-1}left(e^{t}
ight) )
b. ( y=cos ^{-1}left(9^{x}
ight) )

Explanation:

Step1: Recall the derivative formula for \(y = \csc^{-1}(u)\)

The derivative of \(y=\csc^{-1}(u)\) with respect to \(t\) is \(\frac{dy}{dt}=-\frac{1}{|u|\sqrt{u^{2}-1}}\cdot\frac{du}{dt}\), where \(u = e^{t}\) and \(\frac{du}{dt}=e^{t}\). Since \(e^{t}>0\) for all real \(t\), \(|u| = u=e^{t}\).

Step2: Substitute \(u = e^{t}\) into the formula

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

Step3: Simplify the expression

Cancel out the \(e^{t}\) terms. The simplified derivative is \(\frac{dy}{dt}=-\frac{1}{\sqrt{e^{2t}-1}}\).

Answer:

\(\frac{dy}{dt}=-\frac{1}{\sqrt{e^{2t}-1}}\)