Question

Find the derivative of $\frac{a}{x^4}-\frac{b}{x^2}+\cos x$ where $a,b$ are fixed non-zero constants.

Answer 1

Let $f\left(x\right)=\frac{a}{x^4}-\frac{b}{x^2}+\cos x$
$\Rightarrow f\left(x\right)=a{x}^{-4}-b{x}^{-2}+\cos x$
Differentiating with respect to $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}(a{x}^{-4}-b{x}^{-2}+\cos x)$
$\Rightarrow f^{\prime }\left(x\right)=\frac{d}{dx}\left(a{x}^{-4}\right)-\frac{d}{dx}\left(b{x}^{-2}\right)+\frac{d}{dx}\left(\cos x\right)$
$\Rightarrow f^{\prime }\left(x\right)=a\cdot (-4)x^{-4-1}-b\cdot (-2)x^{-2-1}+(-\sin x)$
$\Rightarrow f^{\prime }\left(x\right)=-4a{x}^{-5}+2b{x}^{-3}-\sin x$
$\therefore f^{\prime }\left(x\right)=-\frac{4a}{x^5}+\frac{2b}{x^3}-\sin x$