Question

Differentiate the function given below w.r.t. $x:$

$(a{x}^2+\cot x)(p+q\cos x)$

Answer 1

Given:$(a{x}^2+\cot x)(p+q\cos x)$

Differentiating with respect to $x,$ we get

$\frac{d}{dx}\left[\right(a{x}^2+\cot x\left)\right(p+q\cos x\left)\right]$

$=(p+q\cos x)\frac{d}{dx}(a{x}^2+\cot x)+$

$(a{x}^2+\cot x)\frac{d}{dx}(p+q\cos x)$

$=(2ax-{\csc }^{2}x)(p+q\cos x)+$

$(a{x}^2+\cot x)(-q\sin x)$

$[\because \begin{array}{c}\frac{d}{dx}\cot x=-{\csc }^{2}x\\ \frac{d}{dx}\cos x=-\sin x,\frac{d}{dx}{x}^n=n{x}^{n-1}\end{array}]$

$=(2ax-{\csc }^{2}x)(p+q\cos x)-$

$(a{x}^2+\cot x)\left(q\sin x\right)$