Question 40 of 71intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 4x+5sin⁡x3x+7cos⁡x\frac{4x + 5 \sin x}{3x + 7 \cos x}

Correct Answer

(4+5cos⁡x)(3x+7cos⁡x)−(4x+5sin⁡x)(3−7sin⁡x)(3x+7cos⁡x)2\frac{(4 + 5\cos x)(3x + 7\cos x) - (4x + 5\sin x)(3 - 7\sin x)}{(3x + 7\cos x)^2}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 26 | (Chapter: Page 37)
For More Understanding

Explanation

This question requires applying the quotient rule for differentiation. The textbook context shows the quotient rule formula: for h(x) = u/v, the derivative is h'(x) = (u'v - uv')/v². We identify u = 4x + 5sin x and v = 3x + 7cos x, then compute their derivatives separately before applying the formula.

Solution Steps

  1. Step 1: Let f(x) = (4x + 5sin x)/(3x + 7cos x). Identify u = 4x + 5sin x and v = 3x + 7cos x.

  2. Step 2: Find u' = d/dx(4x + 5sin x) = 4 + 5cos x

  3. Step 3: Find v' = d/dx(3x + 7cos x) = 3 - 7sin x

  4. Step 4: Apply quotient rule: f'(x) = (u'v - uv')/v²

  5. Step 5: f'(x) = [(4 + 5cos x)(3x + 7cos x) - (4x + 5sin x)(3 - 7sin x)]/(3x + 7cos x)²

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