Theory question 111
Differentiation(a) If y = (2x + 3)7 + (x − 1)/(2x − 1), find the value of dy/dx when x = −1.
(b) Using the substitution, u = x + 2, evaluate ∫12 (x − 1)/(x + 2)4 dx
Show the model answer
Model answer
(a) dy/dx = 14(2x + 3)6 + [(2x − 1) − 2(x − 1)]/(2x − 1)2 = 14(2x + 3)6 + 1/(2x − 1)2
At x = −1: 14 + 1/9 = 127/9 (14 1/9)
(b) x − 1 = u − 3, dx = du, limits u = 3 to 4
∫(u−3 − 3u−4)du = [−1/(2u2) + 1/u3] from 3 to 4
= (−1/32 + 1/64) − (−1/18 + 1/27) = −1/64 + 1/54 = 5/1728
Explanation
Chain rule for the power, quotient rule for the fraction; change the limits when you substitute.