Find the derivative of the following function: (4x + 5 sin x) / (3x + 7 cos x)

Solution:

The given function is f(x) = (4x + 5 sin x) / (3x + 7 cos x)

Its derivative using the quotient rule is,

d/dx (f(x)) = [ (3x + 7 cos x) d/dx (4x + 5 sin x) - (4x + 5 sin x) d/dx (3x + 7 cos x) ] / (3x + 7 cos x)²

= [ (3x + 7 cos x) (4 + 5 cos x) - (4x + 5 sin x) (3 - 7 sin x) ] / (3x + 7 cos x)² (using derivative formulas)

= [ 12x + 15x cos x + 28 cos x + 35 cos²x - 12x + 28x sin x - 15 sin x + 35 sin²x ] / (3x + 7 cos x)² 

= [ 35 (cos²x + sin²x) + 15x cos x + 28x sin x - 15 sin x] / (3x + 7 cos x)² 

= [ 35 + 15x cos x + 28x sin x - 15 sin x] / (3x + 7 cos x)²   [As cos²x + sin²x = 1]

NCERT Solutions Class 11 Maths Chapter 13 Exercise ME Question 26

Find the derivative of the following function: (4x + 5 sin x) / (3x + 7 cos x)

Summary:

The derivative of the given function (4x + 5 sin x) / (3x + 7 cos x) is [ 35 + 15x cos x + 28x sin x - 15 sin x] / (3x + 7 cos x)²