Question 59 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): sin⁡nx\sin^n x

Correct Answer

n sinn−1sin^{n-1} x cos x

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 19 | (Page Page 37)
For More Understanding

Explanation

The question asks for the derivative of sinnsin^{n} x. Using the chain rule, we treat sin x as the inner function. The derivative of unu^{n} is n un−1u^{n-1}, and the derivative of sin x is cos x. Combining these: d/dx(sinnsin^{n} x) = n sinn−1sin^{n-1} x · cos x. This follows the pattern shown in Example 18 where the derivative of sin² x was computed as 2 sin x cos x.

Solution Steps

  1. Step 1: Let u = sin x, so sinnsin^{n} x = unu^{n}

  2. Step 2: Apply chain rule: d/dx(unu^{n}) = n un−1u^{n-1} · du/dx

  3. Step 3: Since du/dx = cos x, we get d/dx(sinnsin^{n} x) = n sinn−1sin^{n-1} x · cos x

PreviousNext