Differentiate (i) `sinx^(3)` , (ii) `sin^(3)x`, (iii) `e^(sinx)`

Explanation

(i) Let `y = sin x^(3)`.
Put `x^(3) = t`, so that `y = sint` and `t = x^(3)`
`:. (dy)/(dt) = cos t` and `(dt)/(dx) = 3x^(2)`.
So, `(dy)/(dx) = ((dy)/(dt) xx (dt)/(dx))`
`= 3x^(2) cos t = 3x^(2) cos x^(3)` , `[:' t = x^(3)]`.
Hence, `d/(dx) (sinx^(3)) = 3x^(2) cos x^(3)`.
(ii) Let `y = sin^(3) x = (sinx)^(3)`.
Put `sinx = t`, so that `y = t^(3)` and `t = sin x`.
`:. d/(dt) = 3t^(2)` and `(dt)/(dx) = cosx`.
Hence, `(dy)/(dx) = ((dy)/(dt) xx (dt)/(dx))`
`= 3t^(2) cos x = 3 sin^(2)x cosx [ :' t = sin x]` .
(ii) Let `y = e^(sinx)`
Put `sin x =t`, so that ` y = e^(t)` and `t = sinx`.
`:. (dy)/(dt) = e^(t)` and `(dt)/(dx) = cosx`.
Hence, `(dy)/(dx) = (dy/(dt) xx (dt)/(dx))`
`= e^(t) cos x = e^(sinx). cosx [ :' t = sin x]`.