Math 131 Section 3 3 Derivatives of Trig Functions Limits to remember sin0 0 so sin x lim x 0 sin x sin 0 sin x cos x 1 lim x 0 x x 0 1 0 x is the difference quotient for f x sinx at x 0 Since the limit as x x approaches 0 is 1 f 0 1 cos0 1 so the 2nd limit above is the limit of the difference quotients for cosx cos x cos 0 x 0 cos x 1 0 as x 0 This shows the derivative of cos x at x 0 is 0 x These limits can also be used with addition formulas for sin and cos to show d d sin x cos x dx cos x sin x dx Other derivatives to know which can be derived from the derivatives of sinx and cosx using the rules we have learned d tan x sec 2 d x dx d sec x sec x tan x dx cot x csc dx 2 x d csc x csc x cot x dx Example Derive the formula for the derivative of tanx using the quotient rule d tan x sin x d cos x dx tan x d sin x dx dx cos x 2 cos x sin x sin x cos sec 2 Example Find the derivative of sin x cos x sin x 2 x x f x sin x 1 cos x d cos x dx cos cos 2 2 x x sin cos 2 x 2 x 1 cos 2 x cos x 1 cos x sin x sin x f x 1 cos x 2 Example Find the derivative of f x cos x cos 2 x sin 1 cos x sec x 1 tan x sec x tan x 1 tan x sec x sec 1 tan x 2 x 2 Example f x 1 1 sin x so f x 0 1 cos x 1 sin x 2 x cos x 1 1 cos x 2 1 1 cos x first by using the quotient rule with f as is 2nd by simplifying f and then using the quotient rule f x 2 2 cos x 1 sin x 2
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