Question 70 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): xsin⁡nx\frac{x}{\sin^n x}

Correct Answer

The derivative is sin⁡nx−nxsin⁡n−1xcos⁡xsin⁡2nx\frac{\sin^n x - nx \sin^{n-1} x \cos x}{\sin^{2n} x} or equivalently sin⁡x−nxcos⁡xsin⁡n+1x\frac{\sin x - nx \cos x}{\sin^{n+1} x}.

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 30 | (Page Page 37)
For More Understanding

Explanation

This question asks for the derivative of the function f(x) = x/sinnsin^{n} x. Using the quotient rule, we differentiate by treating x as the numerator and sinnsin^{n} x as the denominator. The derivative of x is 1, and the derivative of sinnsin^{n} x requires the chain rule, giving n·sin^(n-1) x·cos x. Applying the quotient rule formula (u'v - uv')/v² yields the final result.

Solution Steps

  1. Step 1: Apply the quotient rule: d/dx(u/v) = (u'v - uv')/v²

  2. Step 2: Let u = x, so u' = 1. Let v = sinnsin^{n} x, so v' = n·sin^(n-1) x·cos x (using chain rule)

  3. Step 3: Substitute into quotient rule: (1·sinnsin^{n} x - x·n·sin^(n-1) x·cos x)/(sinnsin^{n} x)²

  4. Step 4: Simplify: = (sinnsin^{n} x - nx·sin^(n-1) x·cos x)/sin^(2n) x = (sin x - nx·cos x)/sin^(n+1) x

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