We will need to use Quotient Rule along with the chain Rule.
Also we need to know the basic derivatives of sec and tan.
[tex] \frac{d}{dt} sec(u) = sec(u) tan(u) \frac{du}{dt} [/tex]
[tex] \frac{d}{dt} tan(u) = sec^2 (u) \frac{du}{dt} [/tex]
We will refer to those later.
Now lets look at the Quotient rule:
[tex](\frac{f}{g})' = \frac{f' g - fg'}{g^2}[/tex]
f(t) = sec(3t) , g(t) = tan(5t)
Find f'(t) and g'(t) using derivatives from top of post.
Note that u = 3t for f(t) and u = 5t for g(t).
[tex]f'(t) = 3sec(3t) tan(3t)[/tex]
[tex]g'(t) = 5 sec^2 (5t)[/tex]
Substituting into quotient formula:
[tex]\frac{3 sec(3t) tan(3t) tan(5t) - 5 sec(3t) sec^2 (5t)}{tan^2 (5t)}[/tex]
