Find the derivative of the function sinh - 1 3 x 4 .

Let, y = sinh - 1 3 x 4 Differentiating w.r.t x , we get d y d x = d d x sinh - 1 3 x 4 = 1 1 + 3 x 4 2 d d x 3 x 4 = 3 4 1 1 + 9 x 2 16 = 3 4 · 16 + 9 x 2 16 ∴ d y d x = 3 9 x 2 + 16

Find the derivative of the function log x 2 + x + 2 x 2 - x + 2 .

Let, y = log x 2 + x + 2 x 2 - x + 2 Differentiating w.r.t x , we get d y d x = d d x log x 2 + x + 2 - log x 2 - x + 2 = 1 x 2 + x + 2 d d x x 2 + x + 2 - 1 x 2 - x + 2 d d x x 2 - x + 2 = 2 x + 1 x 2 + x + 2 - 2 x - 1 x 2 - x + 2 = ( 2 x + 1 ) x 2 - x + 2 - ( 2 x - 1 ) x 2 + x + 2 x 2 + x + 2 x 2 - x + 2 = - 2 x 3 - 2 x 2 - 4 x + x 2 + x + 2 x 2 + 2 2 - x 2 ∴ d y d x = 4 - 2 x 2 x 4 + 3 x 2 + 4

Find the derivative of the function cos x sin x + cos x .

Let, y = cos x sin x + cos x Differentiating w.r.t x , we get d y d x = d d x cos x sin x + cos x ⇒ d y d x = sin x + cos x d d x cos x - cos x d d x sin x + cos x ( sin x + cos x ) 2 = ( sin x + cos x ) ( - sin x ) - cos x ( cos x - sin x ) ( sin x + cos x ) 2 = - sin 2 x - sin x cos x - cos 2 x + sin x cos x ( sin x + cos x ) 2 = - 1 sin 2 x + cos 2 x + 2 sin x cos x ∴ d y d x = - 1 1 + sin 2 x

Find the derivative of the function x 1 + x 2 1 - x 2 .

Let, y = x 1 + x 2 1 - x 2 ⇒ y = x + x 3 1 - x 2 Differentiating w.r.t x , we get d y d x = d d x x + x 3 1 - x 2 = 1 - x 2 d d x x + x 3 - x + x 3 d d x 1 - x 2 1 - x 2 = 1 - x 2 1 + 3 x 2 - x + x 3 1 2 1 - x 2 - 2 x 1 x 2 = 1 - x 2 1 + 3 x 2 + x x + x 3 1 - x 2 3 2 = 1 + 3 x 2 - x 2 - 3 x 4 + x 2 + x 4 1 - x 2 3 2 ∴ d y d x = 1 + 3 x 2 - 2 x 4 1 - x 2 3 2