NCERT Solutions
Class 11 Maths
Limits and Derivatives
Ex.13.2 Q.7
For some constants a and b, find the derivative of
(1) (x – a) (x – b)
(2) (ax2 + b)2
(3) (x - a) ÷ (x - b)
(1) Let f(x) = (x – a) (x – b)
=> f(x) = x2 – ax – bx + ab
Now, 
= d (x2 – ax – bx + ab) ÷ dx
=> f’(x) = {d(x2) ÷ dx} – {d(ax) ÷ dx} – {d(bx) ÷ dx} + {d(ab) ÷ dx}
=> f’(x) = 2x – a – b + 0
=> f’(x) = 2x – (a + b)
(2) Let f(x) = (ax2 + b)2
=> f(x) = a2 x4 + b2 + 2abx2
Now, 
= d (a2 x4 + b2 + 2abx2) ÷ dx
=> f’(x) = {d (a2 x4) ÷ dx} + {d(b2) ÷ dx} + {d(2abx2) ÷ dx}
=> f’(x) = 4a2 x3 + 0 + 2ab × 2x
=> f’(x) = 4a2 x3 + 4abx
=> f’(x) = 4ax (ax2 + b)
(3) Let f(x) = (x - a) ÷ (x - b)
Now, = d {(x - a) ÷ (x - b)} ÷ dx
=> f’(x) = {(x - b) × (d (x - a) ÷ dx) - (x - a) × (d (x - b) ÷ dx)} ÷ (x - b)2
[By quotient rule]
=> f’(x) = {(x - b) × 1 - (x - a) × 1} ÷ (x - b)2
=> f’(x) = (x - b - x + a) ÷ (x - b)2
=> f’(x) = (a – b) ÷ (x - b)2