Find the derivative of f x = ( x + 1 ) ( x - 2 ) x with respect to x .

The given function is f x = ( x + 1 ) ( x - 2 ) x ⇒ f x = x 2 - x - 2 x Differentiating both sides with respect to x , we get f ' x = x · d d x x 2 - x - 2 - x 2 - x - 2 · d d x [ x ] ( x ) 2 ⇒ f ' x = 2 x - 1 x - x 2 - x - 2 2 x x Taking LCM and further simplifying, we get f ' x = ( 2 x − 1 ) ( 2 x ) − ( x 2 − x − 2 ) 2 x x ∴ f ' x = 3 x 2 - x + 2 2 x 3 2

Find the derivative of f x = ( x + 1 ) ( x - 2 ) x with respect to x .

The given function is f x = ( x + 1 ) ( x - 2 ) x ⇒ f x = x 2 - x - 2 x Differentiating both sides with respect to x , we get f ' x = x · d d x x 2 - x - 2 - x 2 - x - 2 · d d x [ x ] ( x ) 2 ⇒ f ' x = 2 x - 1 x - x 2 - x - 2 2 x x Taking LCM and further simplifying, we get f ' x = ( 2 x − 1 ) ( 2 x ) − ( x 2 − x − 2 ) 2 x x ∴ f ' x = 3 x 2 - x + 2 2 x 3 2

Find the derivative of f x = ( x + 5 ) ( 2 x 2 - 1 ) x with respect to x .

The given function is f x = ( x + 5 ) ( 2 x 2 - 1 ) x ⇒ f x = 2 x 3 + 10 x 2 - x - 5 x Differentiating both sides with respect to x , we get f ' x = d d x 2 x 3 + 10 x 2 - x - 5 · x - 2 x 3 + 10 x 2 - x - 5 · d d x [ x ] x 2 ⇒ f ' x = 2 · d d x x 3 + 10 · d d x x 2 - d d x [ x ] + d d x [ - 5 ] x - 2 x 3 + 10 x 2 - x - 5 · 1 x 2 ⇒ f ' x = 2 · 3 x 2 + 10 · 2 x - 1 + 0 x - 2 x 3 - 10 x 2 + x + 5 x 2 ⇒ f ' x = - 2 x 3 + x 6 x 2 + 20 x - 1 - 10 x 2 + x + 5 x 2 ⇒ f ' x = 4 x 3 + 10 x 2 + 5 x 2 as the required derivative.