Math 220 October 2 I. Find f 0 (x) 1. f (x) = 6x2 + 5 + 1/x + √ 5 x+ 3x+1 √ 3x 2. f (x) = (x + 2)3 3. f (x) = (x+1)2 x4 4. f (x) = ex+3 + ex−3 II. Differentiate 1. f (x) = x2 ex 2. f (x) = ex 2−ex √ 3. f (x) = (ex + x2 )( x + ex ) 4. f (x) = x+1 x4 +x−1 5. f (x) = x2 −1 x 6. f (x) = x2 −1 x+3 7. f (u) = 1 c+u 8. f (x) = x5/3 (x + kex+k ) 9. f (x) = ax+b cx+d 10. f (x) = (x3 ex )(x5 + 1)(x + 3) 11. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x) 1 √ 3 x cos(x) √ 13. f (x) = (3x)2 + 3x 12. f (x) = 14. f (x) = 2 sec(x) − csc(x) 15. f (θ) = eθ (cot(θ) − θ) 16. f (x) = sin(x) cos(x) 17. f (x) = sin(x) cos(x) tan(x) 18. f (x) = sin(x) cos(x)x 19. f (x) = sin(x) cos(x) tan(x) 20. f (x) = sin(x) cos(x) x 21. f (x) = sec2 (x) 22. f (x) = sin2 (x) + cos2 (x) 23. f (x) = x2 cos(x) tan(x) x+sin(x) 24. f (x) = x+cos(x) x−sin(x) Find an equation of the tangent line to the curve at the given point. 1. y = ex cos(x), 2. y = x + tan(x), (0, 1) (π, π) 2 I. Find f 0 (x) √ √ 1. f (x) = 6x2 + 5 + 1/x + 5 x + 3x+1 3x Solution: f (x) = 6x2 + 5 + x−1 + x1/5 + 3x2/3 + x−1/3 f 0 (x) = 12x + 0 − x−2 + (1/5)x−4/5 + 2x−1/3 + (−1/3)x−4/3 2. f (x) = (x + 2)3 Solution: f (x) = x3 + 6x2 + 12x + 8 f 0 (x) = 3x2 + 12x + 12 2 3. f (x) = (x+1) x4 Solution : 2 2 = xx4 + 2x + x14 = x−2 + 2x−3 + x−4 f (x) = x +2x+1 x4 x4 f 0 (x) = −2x−3 − 6x−4 − 4x−5 4. f (x) = ex+3 + ex−3 Solution: f (x) = ex e3 + ex e−3 f 0 (x) = ex e3 + ex e−3 = ex+3 + ex−3 II. Differentiate 1. f (x) = x2 ex Solution: f 0 (x) = 2xex + x2 ex x e 2. f (x) = 2−e x Solution: 3 f 0 (x) = = = = = (ex )0 (2 − ex ) − ex (2 − ex )0 (2 − ex )2 ex (2 − ex ) − ex (−ex ) (2 − ex )2 ex (2 − ex ) + ex (ex ) (2 − ex )2 2ex − e2x + e2x (2 − ex )2 2ex (2 − ex )2 √ 3. f (x) = (ex + x2 )( x + ex ) Solution: √ √ f 0 (x) = (ex + x2 )0 ( x + ex ) + (ex + x2 )( x + ex )0 √ 1 = (ex + 2x)( x + ex ) + (ex + x2 )( √ + ex ) 2 x 4. f (x) = x4x+1 +x−1 Solution: (x + 1)0 (x4 + x − 1) − (x + 1)(x4 + x − 1)0 (x4 + x − 1)2 (x4 + x − 1) − (x + 1)(4x3 + 1) = (x4 + x − 1)2 f 0 (x) = 2 5. f (x) = x x−1 Solution: f (x) = x2 1 − = x − x−1 x x 4 f 0 (x) = 1 + x−2 2 −1 6. f (x) = xx+3 Solution: (x2 − 1)0 (x + 3) − (x2 − 1)(x + 3)0 f (x) = (x + 3)2 2x(x + 3) − (x2 − 1) = (x + 3)2 x2 + 6x + 1 = (x + 3)2 0 1 7. f (u) = c+u Solution: 10 (c + u) − 1(c + u)0 (c + u)2 0 − 1(0 + 1) = (c + u)2 −1 = (c + u)2 f 0 (x) = 8. f (x) = x5/3 (x + kex+k ) Solution: f 0 (x) = (x5/3 )0 (x + kex+k ) + x5/3 (x + kex ek )0 = (5/3)x2/3 (x + kex+k ) + x5/3 (1 + kex ek ) = (5/3)x2/3 (x + kex+k ) + x5/3 (1 + kex+k ) 5 9. f (x) = ax+b cx+d Solution: (ax + b)0 (cx + d) − (ax + b)(cx + d)0 (cx + d)2 a(cx + d) − (ax + b)c = (cx + d)2 acx + ad − acx − bc = (cx + d)2 ad − bc = (cx + d)2 f 0 (x) = 10. f (x) = (x3 ex )(x5 + 1)(x + 3) Solution: f 0 (x) = (x3 ex )0 [(x5 + 1)(x + 3)] + (x3 ex )[(x5 + 1)(x + 3)]0 = (3x2 ex + x3 ex )[(x5 + 1)(x + 3)] + (x3 ex )[(x5 + 1)0 (x + 3) + (x5 + 1)(x + 3)0 ] = (3x2 ex + x3 ex )[(x5 + 1)(x + 3)] + (x3 ex )[5x4 (x + 3) + (x5 + 1)] = (3x2 ex + x3 ex )[(x5 + 1)(x + 3)] + (x3 ex )[6x5 + 15x4 + 1)] 11. f (x) = sin(x) + cos(x) + tan(x) + csc(x) + sec(x) + cot(x) Solution: f 0 (x) = cos(x) − sin(x) + sec2 (x) − csc(x) cot(x) + sec(x) tan(x) − csc2 (x) √ 12. f (x) = 3 x cos(x) Solution: 6 √ √ f 0 (x) = ( 3 x)0 cos(x) + 3 x(cos(x))0 = (1/3)x−2/3 cos(x) + x1/3 (− sin(x)) = (1/3)x−2/3 cos(x) − x1/3 (sin(x)) 13. f (x) = (3x)2 + Solution: √ 3x f (x) = 9x2 + √ √ 3 x √ 1 3 √ 2 x √ 3 = 18x + √ 2 x f 0 (x) = 18x + 14. f (x) = 2 sec(x) − csc(x) Solution: f 0 (x) = 2 sec(x) tan(x) + csc(x) cot(x) 15. f (θ) = eθ (cot(θ) − θ) Solution: f 0 (x) = (eθ )0 (cot(θ) − θ) + eθ (cot(θ) − θ)0 = eθ (cot(θ) − θ) + eθ (− csc2 (θ) − 1) 16. f (x) = sin(x) cos(x) Solution: 7 f 0 (x) = (sin(x))0 cos(x) + sin(x)(cos(x))0 = cos(x) cos(x) − sin(x) sin(x) = cos2 (x) − sin2 (x) 17. f (x) = sin(x) cos(x) tan(x) Solution: f (x) = sin(x) cos(x) sin(x) = sin(x) sin(x) = sin2 (x) cos(x) f 0 (x) = (sin(x))0 sin(x) + sin(x)(sin(x))0 = cos(x) sin(x) + sin(x) cos(x) = 2 cos(x) sin(x) 18. f (x) = sin(x) cos(x)x Solution: f 0 (x) = (sin(x))0 (cos(x)x) + sin(x)(cos(x)x)0 = cos(x)(cos(x)x) + sin(x)(− sin(x)x + cos(x)) = cos2 (x)x − sin2 (x)x + sin(x) cos(x) 19. f (x) = sin(x) cos(x) tan(x) Solution: f (x) = sin(x) cos(x) sin(x) cos(x) = cos2 (x) f 0 (x) = (cos(x))0 cos(x) + cos(x)(cos(x))0 = − sin(x) cos(x) − cos(x) sin(x) = −2 sin(x) cos(x) 8 20. f (x) = sin(x) cos(x) x Solution: (sin(x) cos(x))0 x − (sin(x) cos(x))x0 x2 0 [(sin(x)) cos(x) + sin(x)(cos(x))0 ]x − (sin(x) cos(x)) = x2 2 2 [cos (x) − sin (x)]x − (sin(x) cos(x)) = x2 f 0 (x) = 21. f (x) = sec2 (x) Solution: f (x) = sec(x) sec(x) f 0 (x) = (sec(x))0 sec(x) + sec(x)(sec(x))0 = (sec(x) tan(x)) sec(x) + sec(x)(sec(x) tan(x)) = 2 sec2 (x) tan(x) 22. f (x) = sin2 (x) + cos2 (x) Solution: f 0 (x) = (sin(x))0 sin(x) + sin(x)(sin(x))0 + (cos(x))0 cos(x) + cos(x)(cos(x))0 = cos(x) sin(x) + sin(x) cos(x) − sin(x) cos(x) − cos(x) sin(x) = cos(x) sin(x) − cos(x) sin(x) + sin(x) cos(x) − sin(x) cos(x) =0 Alternative solution: f (x) = sin2 (x) + cos2 (x) = 1 f 0 (x) = 0 9 2 23. f (x) = x Solution: f 0 (x) = = = = = = = cos(x) tan(x) x+sin(x) [x2 cos(x) tan(x)]0 (x + sin(x)) − [x2 cos(x) tan(x)](x + sin(x))0 (x + sin(x))2 [2x cos(x) tan(x) + x2 (cos(x) tan(x))0 ](x + sin(x)) − [x2 cos(x) tan(x)](1 − cos(x)) (x + sin(x))2 [2x cos(x) tan(x) + x2 (− sin(x) tan(x) + cos(x) sec2 (x))](x + sin(x)) (x + sin(x))2 [x2 cos(x) tan(x)](1 − cos(x)) − (x + sin(x))2 [2x cos(x) tan(x) + x2 (− sin2 (x) sec(x) + sec(x))](x + sin(x)) (x + sin(x))2 [x2 cos(x) tan(x)](1 − cos(x)) − (x + sin(x))2 [2x cos(x) tan(x) + x2 (sec(x)(1 − sin2 (x))](x + sin(x)) (x + sin(x))2 [x2 cos(x) tan(x)](1 − cos(x)) − (x + sin(x))2 [2x cos(x) tan(x) + x2 (sec(x)(cos2 (x))](x + sin(x)) (x + sin(x))2 [x2 cos(x) tan(x)](1 − cos(x)) − (x + sin(x))2 [2x cos(x) tan(x) + x2 (cos(x))](x + sin(x)) − [x2 cos(x) tan(x)](1 − cos(x)) (x + sin(x))2 24. f (x) = x+cos(x) x−sin(x) Solution: 10 (x + cos(x))0 (x − sin(x)) − (x + cos(x))(x − sin(x))0 (x − sin(x))2 (1 − sin(x))(x − sin(x)) − (x + cos(x))(1 − cos(x)) = (x − sin(x))2 f 0 (x) = = Find an equation of the tangent line to the curve at the given point. 1. y = ex cos(x), (0, 1) Solution: y 0 = ex cos(x) − ex sin(x) m = y 0 (0) = e0 cos(0) − e0 sin(0) = 1 − 0 = 1 Tangent line: (y − 1) = 1(x − 0) y =x+1 2. y = x + tan(x), (π, π) Solution: y 0 = 1 + sec2 (x) m = y 0 (π) = 1 + sec2 (π) = 1 + 1 = 2 Tangent line: (y − π) = 2(x − π) y = 2x − π 11