Question:
Find the derivative of the following functions (it is to be understood that $a, b, c, d, p, q, r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$
Solution:
Let $f(x)=\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$
$f^{\prime}(x)=\frac{d}{d x}\left(\frac{a}{x^{4}}\right)-\frac{d}{d x}\left(\frac{b}{x^{2}}\right)+\frac{d}{d x}(\cos x)$
$=a \frac{d}{d x}\left(x^{-4}\right)-b \frac{d}{d x}\left(x^{-2}\right)+\frac{d}{d x}(\cos x)$
$=a\left(-4 x^{-5}\right)-b\left(-2 x^{-3}\right)+(-\sin x) \quad\left[\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\right.$ and $\left.\frac{d}{d x}(\cos x)=-\sin x\right]$
$=\frac{-4 a}{x^{5}}+\frac{2 b}{x^{3}}-\sin x$