Question

Find the derivative of the following function (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):

sinⁿ x

Answer 1

Let y = sinⁿ x.

Accordingly, for n = 1, y = sin x

∴ `(dy)/(dx) = cos x` i.e. `(dy)/(dx) = sin x = cos x`

For n = 2, y = sin² x

∴ `(dy)/(dx) = (d)/(dx) (sin x sin x)`

= (sin x)' sinx + sin x (sin x)'      [By Leibnitz product rule]

= cos x sin x + sin x cos x

= 2 sin x cos x    ...(1)

For n = 3, y = sin³ x

∴ `(dy)/(dx) = (d)/(dx) (sin x sin^2 x)`

= (sin x)' sinx² + sin x (sin² x)       [By Leibnitz product rule]

= cos x sin² x + sin x (2 sin x cos x)     [Using (1)]

= cos x sin² x 2 sin² x cos x

= 3 sin² x cos x

We assert that `d/dx (sin ^n x) = n sin ^(n - 1) x cos x`

Let our assertion be true for n = k.

i.e., `d/dx (sin ^k x) = k sin ^((k - 1)) x cos x`       ...(2)

Consider

`d/dx (sin^(k + 1) x)` = `d/dx (sin x sin^k x)`      

= (sin x)' sinx^k x + sin x (sin^k x)                   [By Leibnitz product rule]

= cos x sin^k x + sin x (k sin^{(k - 1)} x cos x)       [Using (2)]

= cos x sin^k x  + k sin^k x cos x

= (k + 1) sin^k x cos x

Thus, our assertion is true for n = k + 1.

Hence, by mathematical induction, `d/dx(sin^n x)`= n sin^{(n - 1)} x cos x