Question 67 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): x2cos⁡(π4)sin⁡x\frac{x^2 \cos \left(\frac{\pi}{4}\right)}{\sin x}

Correct Answer

The derivative is cos⁡(π4)(2xsin⁡x−x2cos⁡x)sin⁡2x\frac{\cos\left(\frac{\pi}{4}\right)(2x\sin x - x^2\cos x)}{\sin^2 x} or equivalently 2xsin⁡x−x2cos⁡x2sin⁡2x\frac{2x\sin x - x^2\cos x}{\sqrt{2}\sin^2 x}.

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 27 | (Page Page 37)
For More Understanding

Explanation

This question asks for the derivative of a quotient function. Following the quotient rule pattern shown in Example 22 of the context, we apply the formula for derivatives of quotients. Since cos(π/4) = 1/2\sqrt{2} is a constant, it can be factored out during differentiation.

Solution Steps

  1. Step 1: Let f(x) = x²cos(π/4)/sin x. Since cos(π/4) is a constant, write f(x) = cos(π/4) · (x²/sin x)

  2. Step 2: Apply the quotient rule: d/dx[u/v] = (u'v - uv')/v² where u = x² and v = sin x

  3. Step 3: Calculate u' = 2x and v' = cos x

  4. Step 4: Substitute: f'(x) = cos(π/4) · [(2x)(sin x) - (x²)(cos x)]/sin²x

  5. Step 5: Simplify to get f'(x) = cos(π/4)(2x sin x - x² cos x)/sin²x = (2x sin x - x² cos x)/(2\sqrt{2} sin²x)

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