Differentiate with respect to x . 2 x e x

Given function is 2 x e x Let y = 2 x e x Now, d y d x = d d x 2 x e x = 2 d d x x e x = 2 x d d x e x + e x d d x x ∵ d d x u v = u d v d x + v d u d x = 2 x × e x + e x × 1 2 x 1 2 - 1 ∵ d d x x n = n x n - 1 , n ∈ R = 2 x × e x + e x × 1 2 x = 2 2 x e x + e x 2 x Hence, the differentiation of the given function with respect to x is 2 x e x + e x x .

Differentiate with respect to x . e 6 x x

Given function is e 6 x x Let y = e 6 x x Now, d y d x = d d x e 6 x x = x d d x e 6 x - e 6 x d d x x x 2 ∵ d d x u v = v d u d x - u d v d x v 2 = x e 6 x × d d x 6 x - e 6 x x 2 ∵ d d x x n = n x n - 1 , n ∈ R = x e 6 x × 6 - e 6 x x 2 = e 6 x 6 x - 1 x 2 Hence, the differentiation of the given function with respect to x is e 6 x 6 x - 1 x 2 .

Differentiate with respect to x . e - 2 x x

Given function is e - 2 x x Let y = e - 2 x x Now, d y d x = d d x e - 2 x x = x d d x e - 2 x - e - 2 x d d x x x 2 ∵ d d x u v = v d u d x - u d v d x v 2 = x × e - 2 x × d d x - 2 x - e - 2 x d d x x x = x × e - 2 x × - 2 - e - 2 x × 1 2 x x ∵ d d x x n = n x n - 1 , n ∈ R = - 2 x × e - 2 x - e - 2 x 2 x x = - 4 x e - 2 x - e - 2 x 2 x 3 2 Hence, the differentiation of the given function with respect to x is - 4 x e - 2 x - e - 2 x 2 x 3 2 .

Differentiate with respect to x . e x - 1 e x + 2

Given function is e x - 1 e x + 2 Let y = e x - 1 e x + 2 Now, d y d x = d d x e x - 1 e x + 2 = e x + 2 d d x e x - 1 - e x - 1 d d x e x + 2 e x + 2 2 ∵ d d x u v = v d u d x - u d v d x v 2 = e x + 2 × e x - e x - 1 × e x e x + 2 2 ∵ d d x e x = e x , and d d x constant = 0 = e 2 x + 2 e x - e 2 x + e x e x + 2 2 = 3 e x e x + 2 2 Hence, the differentiation of the given function with respect to x is 3 e x e x + 2 2 .

Differentiate with respect to x . x e 3 x + e 6 x 2

Given function is x e 3 x + e 6 x 2 Let y = x e 3 x + e 6 x 2 Now, d y d x = d d x x e 3 x + e 6 x 2 = d d x x e 3 x + 1 2 d d x e 6 x = x d d x e 3 x + e 3 x d d x x + 1 2 × e 6 x × d d x 6 x ∵ d d x u v = u d v d x + v d u d x = 3 x e 3 x + e 3 x + 1 2 × 6 e 6 x ∵ d d x x n = n x n - 1 = 3 x e 3 x + e 3 x + 3 e 6 x Hence, the differentiation of the given function with respect to x is 3 x e 3 x + e 3 x + 3 e 6 x .

Differentiate with respect to x . x 2 e x - x e x + 2

Given function is x 2 e x - x e x + 2 Let y = x 2 e x - x e x + 2 Now, d y d x = d d x x 2 e x - x e x + 2 = e x + 2 d d x x 2 e x - x - x 2 e x - x d d x e x + 2 e x + 2 2 ∵ d d x u v = v d u d x - u d v d x v 2 = e x + 2 d d x x 2 e x - d d x x - x 2 e x - x d d x e x + d d x 2 e x + 2 2 = e x + 2 e x d d x x 2 + x 2 d d x e x - d d x x - x 2 e x - x d d x e x + d d x 2 e x + 2 2 ∵ d d x u v = u d v d x + v d u d x = e x + 2 e x × 2 x + x 2 × e x - 1 - x 2 e x - x e x + 0 e x + 2 2 ∵ d d x x n = n x n - 1 , n ∈ R = e x + 2 2 x e x + x 2 e x - 1 - x 2 e x - x e x e x + 2 2 = 2 x e 2 x + x 2 e 2 x - e x + 4 x e x + 2 x 2 e x - 2 - x 2 e 2 x + x e x e x + 2 2 = 2 x 2 e x + 2 x e 2 x + 5 x e x - e x - 2 e x + 2 2 Hence, the differentiation of the given function with respect to x is 2 x 2 e x + 2 x e 2 x + 5 x e x - e x - 2 e x + 2 2 .