Question

>
If y = x (x-1)(x - 2) then
dy
dx
is
a) 3x2 - 6x +2
b) -6x + 2
c) 3x² + 2​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:\dfrac{dy}{dx}\:=\:3x^2\:-\:6x\:+\:2\:}}}$

Option a) 3x² - 6x + 2

Step-by-step-explanation:

We have given a function.

We have to find the derivative of that function.

The given function is

$\displaystyle{\sf\:y\:=\:x\:(\:x\:-\:1\:)\:(\:x\:-\:2\:)}$

$\displaystyle{\implies\sf\:y\:=\:x\:[\:x\:(\:x\:-\:2\:)\:-\:1\:(\:x\:-\:2\:)\:]}$

$\displaystyle{\implies\sf\:y\:=\:x\:(\:x^2\:-\:2x\:-\:x\:+\:2\:)}$

$\displaystyle{\implies\sf\:y\:=\:x\:(\:x^2\:-\:3x\:+\:2\:)}$

$\displaystyle{\implies\sf\:y\:=\:x^3\:-\:3x^2\:+\:2x}$

Differentiating both sides w.r.t. x, we get,

$\displaystyle{\sf\:\dfrac{d}{dx}\:(\:y\:)\:=\:\dfrac{d}{dx}\:(\:x^3\:-\:3x^2\:+\:2x\:)}$

$\displaystyle{\implies\sf\:\dfrac{dy}{dx}\:=\:\dfrac{d}{dx}\:(\:x^3\:)\:-\:\dfrac{d}{dx}\:(\:3x^2\:)\:+\:\dfrac{d}{dx}\:(\:2x\:)}$

We know that,

$\displaystyle{\boxed{\pink{\sf\:\dfrac{d}{dx}\:(\:x^n\:)\:=\:n\:x^{n\:-\:1}\:}}}$

$\displaystyle{\boxed{\blue{\sf\:\dfrac{d}{dx}\:(\:k\:y\:)\:=\:k\:\dfrac{d}{dx}\:(\:y\:)\:}}}$

$\displaystyle{\implies\sf\:\dfrac{dy}{dx}\:=\:3\:x^{3\:-\:1}\:-\:3\:\dfrac{d}{dx}\:(\:x^2\:)\:+\:2\:\dfrac{d}{dx}\:(\:x\:)}$

$\displaystyle{\implies\sf\:\dfrac{dy}{dx}\:=\:3x^2\:-\:3\:\times\:2\:x^{2\:-\:1}\:+\:2\:\times\:1}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\dfrac{dy}{dx}\:=\:3x^2\:-\:6x\:+\:2\:}}}}$