Find the derivative of the function f x = ( x + 1 ) x 2 - 4 x + 2 x > 0 .

Given, f x = ( x + 1 ) x 2 - 4 x + 2 x > 0 Differentiating w.r.t x , we get Using the product rule, d d x f x g x = f ' x g x + f x g ' x . f ' x = ( x + 1 ) d d x x 2 - 4 x + 2 + x 2 - 4 x + 2 d d x ( x + 1 ) ∴ f ' x = x + 1 2 x - 4 + x 2 - 4 x + 2 2 x

Find the derivative of the function f x = ( x + 1 ) x 2 - 4 x + 2 x > 0 .

Given, f x = ( x + 1 ) x 2 - 4 x + 2 x > 0 Differentiating w.r.t x , we get Using the product rule, d d x f x g x = f ' x g x + f x g ' x . f ' x = ( x + 1 ) d d x x 2 - 4 x + 2 + x 2 - 4 x + 2 d d x ( x + 1 ) ∴ f ' x = x + 1 2 x - 4 + x 2 - 4 x + 2 2 x

Find the derivative of the function f x = p x 2 + q x + r a x + b ; | a | + | b | ≠ 0 .

Given, f x = p x 2 + q x + r a x + b ; | a | + | b | ≠ 0 Let, y = p x 2 + q x + r a x + b Differentiating w.r.t x , we get d y d x = - p x 2 + q x + r d d x a x + b ( a x + b ) 2 = ( a x + b ) ( 2 p x + q ) - p x 2 + q x + r · a ( a x + b ) 2 = 2 a p x 2 + 2 b p x + a q x + b q - a p x 2 - a q x - a r ( a x + b ) 2 ∴ d y d x = a p x 2 + 2 b p x + ( b q - a r ) ( a x + b ) 2

Find the derivative of the function f x = 1 a x 2 + b x + c ; | a | + | b | + | c | ≠ 0 .

Given, f x = 1 a x 2 + b x + c ; | a | + | b | + | c | ≠ 0 Let, y = 1 a x 2 + b x + c Differentiating w.r.t x , we get d y d x = - 1 a x 2 + b x + c 2 d d x a x 2 + b x + c ∴ d y d x = - ( 2 a x + b ) a x 2 + b x + c 2