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c Dr Oksana Shatalov, Fall 2013 1 Section 3.4: Derivatives of Trigonometric Functions It is important to remember that everything for six trigonometric functions (sin x, cos x, tan x, cot x, csc x, sec x) will be done in radians. EXAMPLE 1. Compute: (a) lim sin x = (b) lim cos x = x→0 THEOREM 2. lim x→0 x→0 sin x = 1, x lim x→0 cos x − 1 = 0. x Proof EXAMPLE 3. Find these limits: sin(5x) x→0 x (a) lim sin(9x) x→0 sin(7x) (b) lim x x→0 sin(4x) (c) lim Conclusion: If a, b 6= 0 then lim x→0 sin(ax) = x , lim x x→0 sin(ax) = , lim sin(ax) x→0 sin(bx) = c Dr Oksana Shatalov, Fall 2013 (d) lim 2 1 x→0 x2 cot2 (3x) cos x − 1 x→0 sin x (e) lim EXAMPLE 4. Find the following derivatives: (a) d sinx = dx Remark Similarly one can get (cos x)0 = − sin x. (b) d tanx = dx Derivatives of Trig Functions (memorize these!) d sinx = dx d cosx = − sin x dx d tanx = dx d cscx = − csc x cot x dx d secx = sec x tan x dx d cotx = − csc2 x dx c Dr Oksana Shatalov, Fall 2013 EXAMPLE 5. Find the derivative of these functions. √ (a) y = cot x + 5 sec x + x x (b) f (x) = cos x 1 + sin x 3