Question

If $y=4x^4+2x^3+\frac{5}{x}+9$, then find $dy/dx$.

Answer 1

$\frac{dy}{dx}=\frac{d}{dx}[4x^4+2x^3+\frac{5}{x}+9]$
(Differentiating both sides w.r.t. $x$)
$=\frac{d}{dx}\left[4x^4\right]+\frac{d}{dx}\left[2x^3\right]+\frac{d}{dx}\left[\frac{5}{x}\right]+\frac{d\left(9\right)}{dx}$ (Using the property of linearity)
$=\frac{d}{dx}\left[4x^4\right]+\frac{d}{dx}\left[2x^3\right]+\frac{d}{dx}\left[\frac{5}{x}\right]$
(As differentiation of constant $=0,\frac{d9}{dx}=0)$
$=4\frac{d}{dx},[x{]}^4+2\frac{d}{dx}[x^3]+5\frac{d}{dx}[x^{-1}]$ (Separating constant coefficients)
$=4\times 4x^3+2\times 3x^2+5(-1)x^{-2}$
$=16x^3+6x^2-\frac{5}{x^2}$.