Question

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): sin ⁿ x

Answer 1

Let y = sinⁿ x.
Accordingly, for n = 1, y = sin x. 
\(∴\frac{dy}{dx}\) = cosx, i.e., \(\frac{d}{dx }\) sin x = cosx
For n = 2, y = sin² x.
\(∴\frac{dy}{dx} =\frac{d}{dx}\) (sin x sin x)
= (sin x)' sin x + 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.
\(∴\frac{dy}{dx}\) =\(\frac{d}{dx}\) (sin x sin² x)
= (sin x)' sin² x + sin x (sin² x)' [By Leibnitz product rule]
=cosxsin² x + sin x(2 sin x cos x) [Using (1)]
= cos xsin²x+2 sin² x cos x
=3sin² x cos.

We assert that \(\frac{d}{dx}\) (sin ⁿ x) n sin^(k-1) x cos x
Let our assertion be true for n = k.
i.e., \(\frac{d}{dx}\) (sin^k) = k sin^(k-1) x cos x ...(2)
Consider
d/dx(sink+1 x) d/dx (sin x sin ^k x)
=(sin x)' sink x+ sin x (sink x)'   [By Leibnitz product rule]
=cos x sin^k x + sin x (k sin^(k-1) x cos x)   [Using (2)]
=(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