How do you differentiate #f(x)=tanx# using the quotient rule?

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Answer 1 · 2015-11-07T17:18:40

#f'(tanx)=sec^2x#

Using the trig identities, recall that #tanx# can be rewritten as #sinx/cosx#.
Now differentiate using quotient rule:

#d/dx(f(x)/g(x)) = (g(x)*f'(x) - f(x)*g'(x))/g(x)^2#

#f'(x)=(cosx*cosx-sinx*-sinx)/(cos^2x)#

#f'(x)=(cos^2x+sin^2x)/cos^2x#

Recall from the trig identities that #cos^2x+sin^2x=1#. Simplify:

#f'(x)=1/cos^2x#

Finally, simplify again with trig identities and the result is:

#f'(x)=sec^2x#