Chapter 2 - The Derivative - 2.3 Introduction To Techniques Of Differentiation - Exercises Set 2.3 - Page 141: 46

(a) $12$ (b) $5040$

(a) $y' = 16x^{3} + 6x^{2}$ $\Rightarrow y'' = 48x^{2} + 12x$ $\Rightarrow y''' = 96x + 12$ $\Rightarrow y'''(0) = 96(0) + 12$ $= 12$ (b) $y = 6x^{-4}$ $\Rightarrow \frac{dy}{dx} = -24x^{-5}$ $\Rightarrow \frac{d^{2}y}{dx^{2}} = 120x^{-6}$ $\Rightarrow \frac{d^{3}y}{dx^{3}} = -720x^{-7}$ $\Rightarrow \frac{d^{4}y}{dx^{4}} = 5040x^{-8}$ $\Rightarrow \frac{d^{4}y}{dx^{4}}\vert_{x=1} = 5040(1)^{-8}$ $= 5040$