Answer
(a) $42x-10$
(b) $24$
(c) $2x^{-3}$
(d) $700x^3-96x$
Work Step by Step
(a) $y = 7x^3-5x^2+x$
$\dfrac{dy}{dx} = 7(3)x^{3-2}-5(2)x^{2-1}+x^{1-1}=21x^2-10x+1$
$\dfrac{d^2y}{dx^2}=21(2)x^{2-1}-10x^{1-1}+0=\boxed{42x-10}$
(b) $y = 12x^2-2x+3$
$\dfrac{dy}{dx} = 12(2)x^{2-1}-2=24x-2$
$\dfrac{d^2y}{dx^2}=24x^{1-1}=\boxed{24}$
(c) $y = \dfrac{x+1}{x} = 1+x^{-1}$
$\dfrac{dy}{dx} = (-1)x^{-1-1} =-x^{-2}$
$\dfrac{d^2y}{dx^2} = (-1)(-2)x^{-2-1} = \boxed{2x^{-3}}$
(d) $y = (5x^2-3)(7x^3+x) = 35x^5+5x^3-21x^3-3x = 35x^5-16x^3-3x$
$\dfrac{dy}{dx} = 35(5)x^{5-1}-16(3)x^{3-1}-3 = 175x^4-48x^2-3$
$\dfrac{d^2y}{dx^2} =175(4)x^{4-1}-48(2)x^{2-1}=\boxed{700x^3-96x}$
