Question 56 of 70intermediate🔧 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): cos⁡x1+sin⁡x\frac{\cos x}{1 + \sin x}

Correct Answer

Let f(x) = cos⁡x1+sin⁡x\frac{\cos x}{1 + \sin x}

Using the quotient rule: f'(x) = (cos⁡x)′(1+sin⁡x)−(cos⁡x)(1+sin⁡x)′(1+sin⁡x)2\frac{(\cos x)'(1 + \sin x) - (\cos x)(1 + \sin x)'}{(1 + \sin x)^2}

= (−sin⁡x)(1+sin⁡x)−(cos⁡x)(cos⁡x)(1+sin⁡x)2\frac{(-\sin x)(1 + \sin x) - (\cos x)(\cos x)}{(1 + \sin x)^2}

= −sin⁡x−sin⁡2x−cos⁡2x(1+sin⁡x)2\frac{-\sin x - \sin^2 x - \cos^2 x}{(1 + \sin x)^2}

Since sin⁡2\sin^2 x + cos⁡2\cos^2 x = 1: = −sin⁡x−1(1+sin⁡x)2=−(1+sin⁡x)(1+sin⁡x)2\frac{-\sin x - 1}{(1 + \sin x)^2} = \frac{-(1 + \sin x)}{(1 + \sin x)^2}

= −11+sin⁡x\frac{-1}{1 + \sin x}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 16 | (Page Page 37)
For More Understanding

Explanation

This question requires applying the quotient rule for differentiation, which is demonstrated multiple times in the textbook context (e.g., finding derivative of cot x = cos x/sin x). The quotient rule states that for f(x) = u/v, the derivative is (u'v - uv')/v². Students must also recall that derivative of cos x is -sin x, derivative of sin x is cos x, and use the fundamental identity sin²x + cos²x = 1 to simplify the final expression.

Solution Steps

  1. Step 1: Identify the function and apply quotient rule formula

  2. Step 2: Differentiate numerator (cos x)' = -sin x and denominator (1 + sin x)' = cos x

  3. Step 3: Substitute into quotient rule and expand numerator

  4. Step 4: Apply identity sin²x + cos²x = 1 to simplify

  5. Step 5: Factor and cancel (1 + sin x) from numerator and denominator

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