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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.

  • Calculus -

    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

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