Answer
(a) $\frac{d^{2}x}{dy^{2}} = 168x^{6}-30x$
(b) $\frac{d^{2}x}{dy^{2}} = 0$
(c) $\frac{d^{2}x}{dy^{2}} = \frac{-4}{5x^{3}}$
(d) $\frac{d^{2}y}{dx^{2}} = 24x^{2} + 6$
Work Step by Step
(a) $\frac{dy}{dx} = 28x^{6}-15x^{2}+2$
$\Rightarrow \frac{d^{2}x}{dy^{2}} = 168x^{6}-30x$
(b) $\frac{dy}{dx} = 3$
$\Rightarrow \frac{d^{2}x}{dy^{2}} = 0$
(c) $\frac{dy}{dx} = \frac{((5x)(\frac{d}{dx}[3x-2])) - ((3x-2)(\frac{d}{dx}[5x]))}{(5x)^{2}}$
$= \frac{(5x)(3) - (3x-2)(5)}{25x^{2}}$
$= \frac{15x - (15x-10)}{25x^{2}}$
$= \frac{10}{25x^{2}}$
$= \frac{2}{5x^{2}}$
$\Rightarrow \frac{d^{2}x}{dy^{2}} =\frac{((5x^{2})(\frac{d}{dx}[2]))-((2)(\frac{d}{dx}[5x^{2}]))}{(5x^{2})^{2}} $
$= \frac{0 - 20x}{25x^{4}}$
$= \frac{-4}{5x^{3}}$
(d) $y = 2x^{4}+3x^{2}-10x+15$
$\Rightarrow \frac{dy}{dx} = 8x^{3} + 6x - 10$
$\Rightarrow \frac{d^{2}y}{dx^{2}} = 24x^{2} + 6 $
