Product Rule
y = −x3 (3x4 − 2)
dy/dx = −x3 ⋅ 12x3 + (3x4 − 2) ⋅ −3x2
= −21x 6 + 6x 2
y = 2 / 2x4 − 5
dy/dx = − 2 ⋅ 8x3 (2x4 − 5)2
= − 16x3 / 4x8 − 20x4 + 25
y = (x3 + 3)5
dy/dx = 5(x3 + 3)4 ⋅ 3x2
= 15x2 (x3 + 3)4
A classmate claims that ( f/g ) ' = f '/ g' for any functions f and g. Show an example that proves your classmate wrong.
Many answers.
Ex: f = 4, g = 2x, − 2 /x2 ≠ 0
f (x) = (2x4 − 3)(x2 + 1)
f '(x) = (2x4 − 3) ⋅ 2x + (x2 + 1) ⋅ 8x3
= 12x5 + 8x3 − 6x
y = 4x3 − 3x2 / 4x5 − 4
dy2/dx = (4x5 − 4)(12x2 − 6x) − (4x3 − 3x2 ) ⋅ 20x4 / (4x5 − 4)2
= −8x7 + 9x6 − 12x + 6x / 4x10 − 8x5 + 4
f (x) = cubic root −2x4 + 5
f '(x) = 1/3 (−2x4 + 5)-2/3 ⋅ −8x3
= −8x3 / 3(−2x4 + 5)2/3
Give a function that requires three applications of the chain rule to differentiate. Then differentiate the function.
Many answers: Ex. y = (((2x + 1)5 + 2)6 + 3)7
dy/dx = 7(((2x + 1)5 + 2)6 + 3)6 ⋅ 6((2x + 1)5+ 2)5 ⋅ 5(2x + 1)4 ⋅ 2