Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CEGEP CHAMPLAIN - ST. LAWRENCE 201-103-RE: Differential Calculus Patrice Camiré Problem Sheet #13 Rules of Differentiation 1. Find the derivative of the following functions. (a) 3x2 − 5x + 2 (d) 3x10 − x9 − x + 1 √ (e) x2 − x + 3 (b) 6x3 + x2 − 8x + 9 6 3 2 5 (c) −x + 7x − 4x − 3 2 (f) x + 4x − 2 (g) 2x2 + 3x + π 2 (h) x100 − x50 + 6x2 − 3 7 √ 11 2 (i) 5x − 3x + x − 98 2. Find the derivative of the following functions. Do not use the product rule or the quotient rule. √ √ √ √ √ 4x2 + x + 3 (e) x x − 2 3 x 4 x + 7 (a) x−3 + 2 x − x2 (h) x2 1 3 8 1 3 (b) + +√ + x 1 4 x x2 x 2 (f) √ − 2 + 8 x2 − 8x + 2 x x x √ (i) x3 x−2 1 x (c) 5 − 2 + √ 2 3 x x x 2x − 5x − 1 (g) 2x3 + x − 9 √ √ √ 4 3 5 x (j) (d) 12 x − 6 x + 20 x x4 3. (i) Multiply the expression and find the derivative. (ii) Find the derivative using the product rule. (d) (x4 + 1)(x3 + 1) (a) (2x + 3)(4x − 1) (g) x(5x + 1)(2x + 3) 2 (b) (5x − 1)(3x − 2) (e) (3x − x + 2)(4x + 1) 2 2 (c) (x + 1)(2x − 1) 3 2 (f) (x − x + 3)(x + x + 1) (h) (2x − 1)(3x + 1)(4x + 1) (i) (x2 + 1)(x + 1)(x − 2) 4. Find the derivative of the following functions. x 2x + 1 3x − 1 (b) 4x + 5 (a) (c) x2 + 1 x+1 (e) x2 − x + 2 x2 + 2x + 1 (g) 5x2 − x + 1 2x + 3 (d) x3 − 2 4x − 1 (f) x3 − 1 x3 + 1 (h) x3 − x + 1 x2 + 1 5. Find the derivative of the following functions. (a) (x2 + 3x + 1)4 (b) (x4 − 2x − 3)3 (c) (x2 + 5)4 + 9 8 (d) 1 − (3x + 1)10 3 p (2x + 1)4 + x p (f) 3 (4x2 + 1)3 − x6 (e) 6. Find the derivative of the following functions. (a) x(3x2 − 4x + 1)4 (c) (x3 + 2x − 1)2 (x3 + 2x + 1)2 (b) (x2 + 1)3 (x2 − 5)4 (d) (5x2 + 3x − 1)3 (x2 − 4x + 8)10 7. (i) Find the derivative of the function. (ii) Find all values of x for which the tangent line to the function is horizontal. (3x + 1)2 (a) x2 + 1 (4x − 1)5 (d) (x + 1)3 (b) (2x − 1)3 (6x + 1)2 (e) (5x + 1)6 (2x + 1)4 (c) (x2 + 2)4 (x2 + 3)8 (f) (x2 − 1)3 (x2 − 9)4 3x + 2 8 (g) 4x + 1 s x2 + 1 (h) x2 − 1 6x − 1 4 (i) 5x − 2 (j) x2 − 3x + 1 x2 − 1 10 Answers (d) 30x9 − 9x8 − 1 (g) 4x + 3 (b) 18x + 2x − 8 (e) 2x − 1 (h) 100x99 − 50x49 + 12x (c) −6x5 + 21x2 − 8x (f) 5x4 + 8x (i) 35x6 − 6x + 1 1. (a) 6x − 5 2 2. (a) − 1 3 + √ − 2x 4 x x (b) − 6 1 − − 4x−3/2 x2 x3 (c) − 2 4 1 + − x−4/3 x3 x5 3 (d) 3x−3/4 − 2x−2/3 + 4x−4/5 3√ 7 (e) x − x−5/12 2 6 5 3/2 8 8 (f) x + 3 − 9 2 x x 1 (g) 2 + 2 x (h) − (i) 6 1 − 3 2 x x 3√ 4 x − √ − x−3/2 2 x (j) − 2 3 36 − + x2 x4 x5 (d) x2 (7x4 + 4x + 3) (g) 30x2 + 34x + 3 (b) 30x − 13 (e) 36x2 − 2x + 7 (h) 72x2 + 4x − 5 (c) 6x2 − 2x + 2 (f) 5x4 + 6x2 + 8x − 1 (i) 4x3 − 3x2 − 2x − 1 3. (a) 16x + 10 4. (a) 1 (2x + 1)2 (c) x2 + 2x − 1 (x + 1)2 (e) 3x2 − 2x − 5 (x2 + 2x + 1)2 (g) 5(2x2 + 6x − 1) (2x + 3)2 (b) 19 (4x + 5)2 (d) 8x3 − 3x2 + 8 (4x − 1)2 (f) 6x2 (x3 + 1)2 (h) x4 + 4x2 − 2x − 1 (x2 + 1)2 (e) 8(2x + 1)3 + 1 p 2 (2x + 1)4 + x 5. (a) 4(2x + 3)(x2 + 3x + 1)3 (b) 6(2x3 − 1)(x4 − 2x − 3)2 (c) 64x(x2 + 5)3 (x2 + 5)4 + 9 7 (d) −90(3x + 1)9 1 − (3x + 1)10 2 (f) 2x 4(4x2 + 1)2 − x4 ((4x2 + 1)3 − x6 )2/3 6. (a) (3x2 − 4x + 1)3 (27x2 − 20x + 1) (b) 2x(x2 + 1)2 (x2 − 5)3 (7x2 − 11) (c) 4x(x2 + 2)(3x2 + 2)(x3 + 2x − 1)(x3 + 2x + 1) (d) (5x2 + 3x − 1)2 (x2 − 4x + 8)9 (130x3 − 251x2 + 64x + 112) 7. (a) (i) 2(3x + 1)(3 − x) (x2 + 1)2 (ii) x = 3, − 1 3 6(2x + 3)(2x − 1)2 (b) (i) (6x + 1)3 1 3 (ii) x = , − 2 2 −8x(x2 + 1)(x2 + 2)3 (c) (i) (x2 + 3)9 (ii) x = 0 (4x − 1)4 (8x + 23) (d) (i) (x + 1)4 1 23 (ii) x = , − 4 8 (e) (i) 2(10x + 11)(5x + 1)5 (2x + 1)5 1 11 (ii) x = − , − 5 10 (f) (i) −2x(x2 + 23)(x2 − 1)2 (x2 − 9)5 (ii) x = 0, 1, −1 (g) (i) −40(3x + 2)7 (4x + 1)9 (ii) x = − (h) (i) √ x2 2 3 −2x + 1(x2 − 1)3/2 (ii) x = 0 (i) (i) −28(6x − 1)3 (5x − 2)5 (ii) x = 1 6 10(x2 − 3x + 1)9 (3x2 − 4x + 3) (x2 − 1)11 √ 3± 5 (ii) x = 2 (j) (i)