For some constants a and b, find the derivative of
(i) (x - a)(x - b)    (ii) (ax² + b)²    (iii) (x - a)/(x - b)

Solution:

(i) Let f (x) = (x - a)(x - b)

Therefore,

f (x) = x² - (a + b) x + ab

d/dx f(x) = d/dx [x² - (a + b) x + ab]

= d/dx (x²) - (a + b) dy/dx (x) + d/dx (ab)

On using derivative formula d/dx (xⁿ) = nx^{n - 1}, we obtain

d/dx f (x) = 2x - (a + b) (1) + 0

= 2x - a - b

(ii) Let f (x) = (ax² + b)²

Therefore,

f (x) = a²x⁴ + 2abx² + b²

= d/dx (a²x⁴ + 2abx² + b²)

= a²d/dx (x⁴) + 2ab d/dx (x²) + d/dx b²

On using theorem d/dx (xⁿ) = nx^{n - 1}, we obtain

f (x) = a² (4x³) + 2ab (2x) + b² (0)

= 4a²x³ + 4abx

= 4ax (ax² + b)

(iii) Let f (x) = (x - a) / (x - b)

Therefore, f (x) = d/dx [ (x - a) / (x - b) ]

By quotient rule,

f (x) = [(x - b) d/dx (x - a) - (x - a) d/dx (x - b)]/(x - b)²

= [(x - b)(1) - (x - a)(1)]/(x - b)²

= (x - b - x + a)/(x - b)²

= (a - b)/(x - b)²

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NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.2 Question 7

For some constants a and b, find the derivative of (i) (x - a)(x - b) (ii) (ax² + b)² (iii) (x - a)/(x - b)

Summary:

The derivatives of the given functions are (i) 2x - a - b (ii) 4ax (ax² + b) (iii) (a - b)/(x - b)²