Question 58 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): sec⁡x−1sec⁡x+1\frac{\sec x - 1}{\sec x + 1}

Correct Answer

Let y = sec⁡x−1sec⁡x+1=1cos⁡x−11cos⁡x+1=1−cos⁡x1+cos⁡x\frac{\sec x - 1}{\sec x + 1} = \frac{\frac{1}{\cos x} - 1}{\frac{1}{\cos x} + 1} = \frac{1 - \cos x}{1 + \cos x}

Using quotient rule:

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

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

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

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 18 | (Page Page 37)
For More Understanding

Explanation

This is a 1-mark question from Exercise 12.2. The student should first simplify the expression by converting sec x to 1/cos x, then apply the quotient rule. The derivative of cos x is -sin x (from the context showing derivative of sin x is cos x, implying derivative of cos x is -sin x). The numerator simplifies to 2sin x after expanding and combining like terms.

Solution Steps

  1. Step 1: Simplify the expression: Convert sec x to 1/cos x and simplify to get (1 - cos x)/(1 + cos x)

  2. Step 2: Apply quotient rule: dy/dx = [f'(x)g(x) - f(x)g'(x)]/[g(x)]² where f(x) = 1 - cos x and g(x) = 1 + cos x

  3. Step 3: Calculate derivatives: f'(x) = sin x and g'(x) = -sin x

  4. Step 4: Substitute and simplify: Result = 2sin x/(1 + cos x)²

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