10. find the derivative.\na. ( y=csc ^{-1}left(e^{t}\right) )\nb. ( y=cos ^{-1}left(9^{x}\right) )

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}})