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MATH 12002 - CALCULUS I §2.3 & §2.4: Derivatives of Trigonometric Functions Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 11 Derivatives of Sine & Cosine Our first two differentiation formulas for the trigonometric functions are d d sin x = cos x and cos x = − sin x. dx dx The first of these is proved in the text; we will prove the second using the definition of derivative. We will also need the angle sum formula for cosine, cos(A + B) = cos A cos B − sin A sin B, and the limits from Equation 6 and Example 11 of §1.4, sin θ cos θ − 1 = 1 and lim = 0. θ→0 θ θ→0 θ lim D.L. White (Kent State University) 2 / 11 Derivatives of Sine & Cosine By definition of the derivative, we have d cos x dx = = = = = = = D.L. White (Kent State University) cos(x + h) − cos x h [cos x cos h − sin x sin h] − cos x lim h→0 h cos x(cos h − 1) − sin x sin h lim h→0 h cos h − 1 sin h lim cos x − sin x h→0 h h cos h − 1 sin h lim cos x − lim sin x h→0 h→0 h h cos h − 1 sin h (cos x) lim − (sin x) lim h→0 h→0 h h (cos x)(0) − (sin x)(1) = − sin x. lim h→0 3 / 11 Derivatives of Other Trigonometric Functions Recall that the other trigonometric functions can be written in terms of sin x and cos x: tan x = sin x , cos x cot x = cos x , sin x sec x = 1 , cos x csc x = 1 . sin x We can use these relations and the derivatives of sin x and cos x to find the derivatives of all of the trigonometric functions. D.L. White (Kent State University) 4 / 11 Derivatives of Other Trigonometric Functions Using the Quotient Rule, we have d d sin x tan x = dx dx cos x d d dx (sin x) cos x − sin x dx (cos x) = cos2 x Hence d dx = (cos x) cos x − sin x(− sin x) cos2 x = cos2 x + sin2 x 1 = = sec2 x. 2 cos x cos2 x tan x = sec2 x. Similarly, D.L. White (Kent State University) d dx cot x = − csc2 x. 5 / 11 Derivatives of Other Trigonometric Functions Again using the Quotient Rule, we have d 1 d sec x = dx dx cos x d d dx (1) cos x − 1 · dx (cos x) = cos2 x Hence d dx = (0) cos x − 1(− sin x) cos2 x = sin x 1 sin x = · = sec x tan x. 2 cos x cos x cos x sec x = sec x tan x. Similarly, D.L. White (Kent State University) d dx csc x = − csc x cot x. 6 / 11 Derivatives of Other Trigonometric Functions We now have Derivatives of Trigonometric Functions d dx sin x = cos x d dx cos x = − sin x d dx tan x = sec2 x d dx cot x = − csc2 x d dx sec x = sec x tan x d dx csc x = − csc x cot x Notes: Derivatives involving cot x and csc x may show up in homework problems on WebAssign occasionally due to randomization, but they will not appear on any exams. The derivative of sec x is the product, sec x tan x = (sec x)(tan x), and not the composite function, sec tan x = sec(tan x). D.L. White (Kent State University) 7 / 11 Examples 1 3 sin x . Find the derivative of f (x) = √ x + 5x 2 By the quotient rule d f 0 (x) = = √ 2 ) − (3 sin x) d (√x + 5x 2 ) (3 sin x) ( x + 5x dx dx √ ( x + 5x 2 )2 √ (3 cos x)( x + 5x 2 ) − (3 sin x)( 21 x −1/2 + 10x) √ . ( x + 5x 2 )2 D.L. White (Kent State University) 8 / 11 Examples 2 Find the derivative of F (x) = 3x 5 tan x. By the product rule, with f (x) = 3x 5 and g (x) = tan x, d d 0 5 5 F (x) = (3x ) (tan x) + (3x ) (tan x) dx dx = 15x 4 tan x + 3x 5 sec2 x. D.L. White (Kent State University) 9 / 11 Examples 3 Find the derivative of F (x) = x 2 sin x cos x. By the general product rule, with f (x) = x 2 , g (x) = sin x, and h(x) = cos x, d 2 d d F 0 (x) = dx x sin x cos x + x 2 dx sin x cos x + x 2 sin x dx cos x = 2x sin x cos x + x 2 cos x cos x + x 2 sin x(− sin x). D.L. White (Kent State University) 10 / 11 Examples 4 Find the derivative of f (x) = sec x tan x . cos x By the quotient rule d 0 f (x) = = d tan x) (cos x) − (sec x tan x) dx (cos x) cos2 x (sec x tan x) tan x + sec x(sec2 x) cos x − (sec x tan x)(− sin x) . cos2 x dx (sec x D.L. White (Kent State University) 11 / 11