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  4. If the derivative of x4(5 sin x-3 cos x) is x3[m x cos x+n x sin x+p sin x+q cos x], then m, n, p and q respectively are

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Q. If the derivative of $x^4(5 \sin x-3 \cos x)$ is $x^3[m x \cos x+n x \sin x+p \sin x+q \cos x]$, then $m, n$, $p$ and $q$ respectively are

Limits and Derivatives

Solution:

Let $y=x^4(5 \sin x-3 \cos x)$
Differentiating $y$ w.r.t. $x$, we get
$\frac{d y}{d x}=x^4 \frac{d}{d x}(5 \sin x-3 \cos x)+(5 \sin x-3 \cos x) \frac{d}{d x}\left(x^4\right)$(by product rule)
$=x^4(5 \cos x+3 \sin x)+(5 \sin x-3 \cos x) 4 x^3$
$=x^3[x(5 \cos x+3 \sin x)+4(5 \sin x-3 \cos x)]$
$=x^3[5 x \cos x+3 x \sin x+20 \sin x-12 \cos x]$