Math 220 February 21 I. Find the derivative. 1. f (x) = 1/x2 + 5x3 + πx2 + x− √ 2 + eπ 2. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x) + ex 3. f (x) = 2x + e3x + 32x 4. f (x) = sin(3x) + (x + 1)2 + e4x+1 + tan(cx) 5. f (x) = √ x2 + 3x + 6 6. f (x) = cos(ex ) + ecos(x) 7. f (x) = sin2 (x) 8. f (x) = e(x 2 +x+x−1 ) 9. f (x) = (x2 + x + 3x4 )−2 10. f (x) = tan(3x + sin(x)) 11. f (x) = p √ sin(x) + sin( x) 12. f (x) = csc(x2 + x) √ 13. f (x) = 4 x+1 1 r 14. f (x) = 15. f (x) = ex + 1 x2 + 1 x2 + 2 x3 + x 7 √ 16. f (x) = x+3 sin(x + 3) q p √ 17. f (x) = x + x + x + 1 18. f (x) = cot(sec(x2 ) + cos(x3 )) 19. f (x) = sin2 (cos2 (x2 + x) + x) 20. 2( x4 f (x) = 3 II. Find the first and second derivatives. 1. f (x) = ex 2. f (x) = x2 + 2x 3. f (x) = sin2 (x) 4. f (x) = cos(αx) sin(βx) 5. f (x) = x2 ex + xex + ex 2 ) ! 1 Solutions I. Find the derivative. 1. f (x) = 1/x2 + 5x3 + πx2 + x− Answer: √ 2 + eπ √ √ f 0 (x) = −2x−3 + 15x2 + 2πx + (− 2)x− 2−1 2. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x) + ex Answer: f 0 (x) = cos(x)−sin(x)+sec2 (x)−csc(x) cot(x)+sec(x) tan(x)−csc2 (x)+ex 3. f (x) = 2x + e3x + 32x f 0 (x) = ln(2)2x + 3e3x + ln(3)32x 2 4. f (x) = sin(3x) + (x + 1)2 + e4x+1 + tan(cx) f 0 (x) = 3 cos(3x) + 2(x + 1) + 4e4x+1 + c sec2 (cx) √ 5. f (x) = x2 + 3x + 6 Answer: 1 f 0 (x) = √ (x2 + 3x + 6)0 2 2 x + 3x + 6 1 = √ (2x + 3) 2 x2 + 3x + 6 2x + 3 = √ 2 x2 + 3x + 6 6. f (x) = cos(ex ) + ecos(x) Answer: 3 f 0 (x) = − sin(ex )(ex )0 + ecos(x) (cos(x))0 = − sin(ex )ex − sin(x)ecos(x) 7. f (x) = sin2 (x) Answer: f 0 (x) = 2(sin(x))(sin(x))0 = 2 sin(x) cos(x) = sin(2x) 8. f (x) = e(x Answer: 2 +x+x−1 ) 2 +x+x−1 ) (x2 + x + x−1 )0 2 +x+x−1 ) (2x + 1 − x−2 ) f 0 (x) = e(x = e(x 9. f (x) = (x2 + x + 3x4 )−2 Answer: f 0 (x) = −2(x2 + x + 3x4 )−3 (x2 + x + 3x4 )0 = −2(x2 + x + 3x4 )−3 (2x + 1 + 12x3 ) 10. f (x) = tan(3x + sin(x)) Answer: f 0 (x) = sec2 (3x + sin(x))(3x + sin(x))0 = sec2 (3x + sin(x))(ln(3)3x + cos(x)) 4 p √ 11. f (x) = sin(x) + sin( x) Answer: √ √ 1 (sin(x))0 + cos( x)( x)0 f 0 (x) = p 2 sin(x) √ cos( x) cos(x) √ + = p 2 x 2 sin(x) 12. f (x) = csc(x2 + x) Answer: f 0 (x) = − csc(x2 + x) cot(x2 + x)(x2 + x)0 = − csc(x2 + x) cot(x2 + x)(2x + x) √ 13. f (x) = 4 Answer: x+1 √ ( x + 1)0 √ 1 = ln(4)4 x+1 ( √ ) 2 x f 0 (x) = ln(4)4 r 14. f (x) = ex + 1 x2 + 1 Answer: 5 √ x+1 1 ex + 1 0 ) f 0 (x) = r x ( 2 e +1 x +1 2 x2 + 1 x 1 (e + 1)0 (x2 + 1) − (ex + 1)(x2 + 1)0 = r x (x2 + 1)2 e +1 2 x2 + 1 x 2 e (x + 1) − (ex + 1)2x 1 = r x (x2 + 1)2 e +1 2 x2 + 1 15. f (x) = x2 + 2 x3 + x 7 Answer: 0 f (x) = 7 x2 + 2 x3 + x 6 x2 + 2 x3 + x 6 0 (x2 + 2)0 (x3 + x) − (x2 + 2)(x3 + x)0 =7 x3 + x 6 2 (2x)(x3 + x) − (x2 + 2)(3x2 + 1) x +2 =7 x3 + x x3 + x x2 + 2 x3 + x √ x+3 sin(x + 3) Answer: 16. f (x) = √ √ ( x + 3)0 sin(x + 3) − ( x + 3)(sin(x + 3))0 f (x) = sin2 (x + 3) √ (1/2)(x + 3)−1/2 sin(x + 3) − ( x + 3)(cos(x + 3)) = sin2 (x + 3) 0 6 q p √ 17. f (x) = x + x + x + 1 Answer: 0 f (x) = = = q 1 p (x + √ 2(x + x + x + 1) x+ √ x + 1)0 1 1+ p (x + √ 2 x+ x+1 1 p √ 2(x + x + x + 1) √ ! x + 1)0 ! 1 p √ 2(x + x + x + 1) 1 1 1+ p (1 + √ ) √ 2 x+1 2 x+ x+1 18. f (x) = cot(sec(x2 ) + cos(x3 )) Answer: f 0 (x) = − csc2 (sec(x2 ) + cos(x3 ))(sec(x2 ) + cos(x3 ))0 = csc2 (sec(x2 ) + cos(x3 ))(sec(x2 ) tan(x2 )2x − sin(x3 )3x2 ) 19. f (x) = sin2 (cos2 (x2 + x) + x) Answer: f 0 (x) = 2 sin(cos2 (x2 + x) + x) (sin(cos2 (x2 + x) + x))0 = 2 sin(cos2 (x2 + x) + x) (cos(cos2 (x2 + x) + x)(cos2 (x2 + x) + x)0 = 2 sin(cos2 (x2 + x) + x) (cos(cos2 (x2 + x) + x)(2 cos(x2 + x)(cos(x2 + x))0 + 1) = 2 sin(cos2 (x2 + x) + x) · (cos cos2 (x2 + x) + x (2 cos(x2 + x)(− sin(x2 + x)(2x + 1)) + 1) 20. 2( f (x) = 3 7 x4 ) ! Answer: 2( x4 f 0 (x) = ln(3)3 ! ) x4 2( ) ! x4 2( ) ! 0 0 4 ln(2)2(x ) x4 = ln(3)3 = ln(3)3 4 2(x ) 4 ln(2)2(x ) 4x3 II. Find the first and second derivatives. 1. f (x) = ex Answer: f 0 (x) = ex f 00 (x) = ex 2. f (x) = x2 + 2x Answer: f 0 (x) = 2x + ln(2)2x f 00 (x) = 2 + (ln(x))2 2x 3. f (x) = sin2 (x) Answer: 8 f 0 (x) = 2 sin(x) cos(x) f 00 (x) = 2 cos(x) cos(x) + 2 sin(x)(− sin(x)) = 2(cos( x) − sin2 (x)) = 2 cos(2x) 4. f (x) = cos(αx) sin(βx) Answer: f 0 (x) = −α sin(αx) sin(βx) + β cos(αx) cos(βx) f 00 (x) = −α2 cos(αx) sin(βx) − αβ sin(αx) cos(βx) − βα sin(αx) cos(βx) − β 2 cos(αx) sin(βx) = −(α2 + β 2 ) cos(αx) sin(βx) − 2αβ sin(αx) cos(βx) 5. f (x) = x2 ex + xex + ex Answer: f 0 (x) = 2xex + x2 ex + ex + xex + ex = x2 ex + 3xex + 2ex f 00 (x) = 2xex + x2 ex + 3ex + 3xex + 2ex 9