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hala718 eNotes educator| Certified Educator

W need to find the derivative of tanx.

We know that tanx = sinx/cosx

Since tan x is a quotient of two functions, then we will use the quotient rule to find the derivative.

We know that if f(x)= u/v , then f'(x)= (u'v - uv')/v^2

==> Let tanx = u/v such that:

u= sinx ==> u' = cosx

v= cosx ==> v' = -sinx

==> (tanx)' = ( cosx*cosx - sinx*-sinx)/cos^2 x

==> (tanx)' = (cos^2x + sin^2 x)/cos^2 x

But we know that cos^2 x + sin^2 x = 1

==> (tanx)' = 1/cos^2 x = (1/cosx)^2

Also we know that sec x = 1/cos x

==> (tanx)' = sec^2 x........

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