Question

y = (x+1)(x+2) find dy/dx​

Answer 1

Answer:

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

Step-by-step-explanation:

We have given a function.

We have to find the derivative of that function.

The given function is

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

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

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

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

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

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

We know that,

$\displaystyle{\boxed{\blue{\sf\:\dfrac{d}{dx}\:\left(\:u\:+\:v\:\right)\:=\:\dfrac{du}{dx}\:+\:\dfrac{dv}{dx}\:}}}$

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

We know that,

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

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

We know that,

$\displaystyle{\boxed{\green{\sf\:\dfrac{d}{dx}\:(\:kx\:)\:=\:k\:\dfrac{d}{dx}\:(\:x\:)\:}}\sf\quad\:k\:is\:constant}$

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

We know that,

$\displaystyle{\boxed{\purple{\sf\:\dfrac{d}{dx}\:(\:k\:)\:=\:0\:}}\sf\quad\:k\:is\:constant}$

$\displaystyle{\implies\sf\:\dfrac{dy}{dx}\:=\:2x\:+\:3\:\times\:1\:+\:0}$

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