Question

1.) Find the derivative of tan (sec x).

2.) Find the derivative if 1/x in four ways, using the limit process, power rule, quotient rule and implicit differentiation.

3.) Show that the derivative of sec^-1 x is 1/(|x|*sqrt(x^2 -1)).

4. Find the derivative of 2^(e^(sin x)).

Thank you in advance.

Answer 1

d/du(tan u) = sec^2 u
so, use that and apply the chain rule:
sec^2(sec x) * secx tanx

I'm sure you can do #2

y = sec^-1 x
sec y = x
secy tany y' = 1
y' = 1/(secy tany)
y' = 1/(x √(x^2-1))

y = 2^(e^(sinx))
y' = ln2 2^(e^(sinx)) * e^(sinx) * cosx