4x +5Sin x÷ 3x+7Cosx

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Solution

Let y = (4x+5Sinx) /(3x+7Cosx)

then dy/dx = {(3x + 7cosx)(4 + 5cosx) - (4x + 5sinx)(3 - 7sinx)}/(3x + 7sinx)²

= {(12x + 15xcosx + 28cosx + 35cos² x) - (12x - 28xsinx + 15sinx - 35sin²x)}/(3x + 7sinx)²

= {12x + 15xcosx + 28cosx + 35cos² x - 12x + 28xsinx - 15sinx + 35sin²x)}/(3x + 7sinx)²

= {15xcosx + 28cosx + 28xsinx -15sinx + 35(cos² x + sin² x)}/(3x + 7sinx)²

= {15xcosx + 28cosx + 28xsinx -15sinx + 35}/(3x + 7sinx)² (since cos² x + sin² x = 1)

**So derivative of (4x+5Sinx) /(3x+7Cosx) = {15xcosx + 28cosx + 28xsinx -15sinx + 35}/(3x + 7sinx)²**

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