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प्रश्न
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
उत्तर
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)' sinxk x + sin x (sink x) [By Leibnitz product rule]
= cos x sink x + sin x (k sin(k - 1) x cos x) [Using (2)]
= cos x sink x + k sink x cos x
= (k + 1) sink 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
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