Question

(-7x^2-5x)cosx differentiate with respect to x​

Answer 1

Question:-

$\displaystyle\sf\dfrac{d}{dx}\:(-7x^2-5x)\:cos\:x$

AnSwEr:-

• use product rule

$\displaystyle\bf\dfrac{d}{dx}(uv)=v\:\dfrac{du}{dx}+u\:\dfrac{dv}{dx}$

• u = -7x² - 5x
• v = cos x

$\displaystyle\sf\implies cos\:x\left(\dfrac{d}{dx}\:-5x-7x^2\right) + (-5x-7x^2)\left(\dfrac{d}{dx}\:cos\:x\right)$

• differentiate the sum term and factor out constants

$\displaystyle\sf = (-5x-7x^2)\left(\dfrac{d}{dx}\:cos\:x\right)+\left[-5\left(\dfrac{d}{dx}\:x-7\:\dfrac{d}{dx}\:x^2\right)\right]cos\:x$

$\displaystyle\sf = (-5x-7x^2)\left(\dfrac{d}{dx}\:cos\:x\right)+cos\:x\:\left(-7\left(\dfrac{d}{dx}\:x^2\right)-1\times 5\right)$

• use power rule

$\displaystyle\bf \dfrac{d}{dx}\:x^n = nx^{n-1}$

• n = 2
• $\displaystyle\sf \dfrac{d}{dx} x^2 = 2x$

$\displaystyle\sf\implies (-5x-7x^2)\left(\dfrac{d}{dx}\:cos\:x\right)+cos\:x\:(-5-7\times 2x)$

• use chain rule

$\displaystyle\bf \dfrac{d}{dx}\:cos\:x = \dfrac{d\:cos\:u}{du}\:\dfrac{du}{dx}$

• u = x
• $\displaystyle\sf \dfrac{d}{du} (cos\:u)=\:-sin\:u$

$\displaystyle\sf\implies (-5-14x)\:cos\:x+(-5x-7x^2)\dfrac{d}{dx}\:x\:sin\:x$

$\displaystyle\boxed{\sf = -(5+14x)\:cos\:x+x(5+7x)\:sin\:x}$