Differentiate the following w.r.t. x : x 2 + x + 1 .

Let y = x 2 + x + 1 Differentiating the above equation with respect to x , d y d x = 1 2 x 2 + x + 1 d d x x 2 + x + 1 {using chain rule of differentiation} d y d x = 1 2 x 2 + x + 1 d d x x 2 + d d x x + d d x 1 d y d x = 1 2 x 2 + x + 1 2 x + 1 .

Differentiate the following w.r.t. x : x 2 + x + 1 .

Let y = x 2 + x + 1 Differentiating the above equation with respect to x , d y d x = 1 2 x 2 + x + 1 d d x x 2 + x + 1 {using chain rule of differentiation} d y d x = 1 2 x 2 + x + 1 d d x x 2 + d d x x + d d x 1 d y d x = 1 2 x 2 + x + 1 2 x + 1 .

Differentiate the following w.r.t. x : 1 a 2 - x 2 .

Let y = 1 a 2 - x 2 = a 2 - x 2 - 1 2 Differentiating above equation with respect to x , d y d x = d d x a 2 - x 2 - 1 2 d y d x = - 1 2 . 1 a 2 - x 2 3 2 d d x a 2 - x 2 {using chain rule of differentiation} d y d x = - 1 2 . 1 a 2 - x 2 3 2 - 2 x = x a 2 - x 2 3 2 .

Differentiate the following w.r.t. x : sin - 1 a + b cos x b + a cos x , b > a > 1 .

Let y = sin - 1 a + b cos x b + a cos x Differentiating above equation with respect to x , ⇒ d y d x = d d x sin - 1 a + b cos x b + a cos x ⇒ d y d x = 1 1 - a + b cos x b + a cos x 2 d d x a + b cos x b + a cos x {using chain rule of differentiation} ⇒ d y d x = b + a cos x b + a cos x 2 - a + b cos x 2 b + a cos x d d x a + b cos x - a + b cos x d d x b + a cos x b + a cos x 2 ⇒ d y d x = 1 b 2 1 - cos 2 x - a 2 1 - cos 2 x b + a cos x - b sin x - a + b cos x - a sin x b + a cos x ⇒ d y d x = 1 b 2 sin 2 x - a 2 sin 2 x - b 2 sin x + a b sin x cos x + a 2 sin x + a b sin x cos x b + a cos x ⇒ d y d x = 1 sin x b 2 - a 2 a 2 sin x - b 2 sin x b + a cos x ⇒ d y d x = - b 2 - a 2 b + a cos x .

Differentiate the following w.r.t. x : log e sin x 2 3 - 1 .

Let y = log e sin x 2 3 - 1 Differentiating the above equation with respect to x , ⇒ d y d x = 1 2 log e sin x 2 3 - 1 d d x log e sin x 2 3 - 1 (using Chain rule of differentiation) ⇒ d y d x = 1 2 log e sin x 2 3 - 1 × 1 sin x 2 3 - 1 × d d x sin x 2 3 - 1 ⇒ d y d x = 1 2 log e sin x 2 3 - 1 × 1 sin x 2 3 - 1 × cos x 2 3 - 1 × d d x x 2 3 - 1 ⇒ d y d x = 1 2 log e sin x 2 3 - 1 × 1 sin x 2 3 - 1 × cos x 2 3 - 1 × 2 x 3 ⇒ d y d x = x cot x 2 3 - 1 3 log e sin x 2 3 - 1

Differentiate the following w.r.t. x : log e x + a 2 + x 2 .

Let y = log e x + a 2 + x 2 Differentiating above equation with respect to x , ⇒ d y d x = d d x log e x + a 2 + x 2 ⇒ d y d x = 1 x + a 2 + x 2 d d x x + a 2 + x 2 (using Chain rule of differentiation) ⇒ d y d x = 1 x + a 2 + x 2 d d x x + d d x a 2 + x 2 ⇒ d y d x = 1 x + a 2 + x 2 1 + 1 2 a 2 + x 2 d d x a 2 + x 2 ⇒ d y d x = 1 x + a 2 + x 2 1 + 1 2 a 2 + x 2 2 x ⇒ d y d x = 1 x + a 2 + x 2 x + a 2 + x 2 a 2 + x 2 ⇒ d y d x = 1 a 2 + x 2 .

Differentiate the following w.r.t. x : log e a + b sin x a - b sin x

Let y = log e a + b sin x a - b sin x Differentiating above equation with respect to x , ⇒ d y d x = d d x log e a + b sin x a - b sin x ⇒ d y d x = 1 a + b sin x a - b sin x d d x a + b sin x a - b sin x (using Chain rule of differentiation) ⇒ d y d x = a - b sin x a + b sin x a - b sin x d d x a + b sin x - a + b sin x d d x a - b sin x a - b sin x 2 ⇒ d y d x = 1 a + b sin x a - b sin x b cos x - a + b sin x - b cos x a - b sin x ⇒ d y d x = 1 a + b sin x a b cos x - b 2 sin x cos x + a b cos x + b 2 sin x cos x a - b sin x ⇒ d y d x = 2 a b cos x a 2 - b 2 sin 2 x ( ∵ x + y x - y = x 2 - y 2 ).