Question

Find the derivative of the following functions (it is to be understood that $a,b,c,d,p,q,r,s$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers) : $x^4(5\sin x-3\cos x)$

Answer 1

Let $f\left(x\right)=x^2(5\sin x-3\cos x)$
Thus by product rule
$f^{\prime }\left(x\right)=x^4\frac{d}{dx}(5\sin x-3\cos x)+(5\sin x-3\cos x)\frac{d}{dx}\left(x^4\right)$
= $x^4[5\frac{d}{dx}\left(\sin x\right)-3\frac{d}{dx}\left(\cos x\right)]+(5\sin x-3\cos x)\frac{d}{dx}\left(x^4\right)$
= $x^4[5\cos x-3(-\sin x)]+(5\sin x-3\cos x)\left(4x^3\right)$
= $x^3[5x\cos x+3x\sin x+20\sin x-12\cos x]$