Differentiation 1 Differentiation 1

Solutions to Exercises

Solution to Exercise 3:

(i)

We have $f(x)=x^{3}$ , and so

$\displaystyle\frac{f(x+h)-f(x)}{h}$ $\displaystyle=\frac{(x+h)^{3}-x^{3}}{h}$

$\displaystyle=\frac{x^{3}+3hx^{2}+3h^{2}x+h^{3}-x^{3}}{h}$

$\displaystyle=\frac{3hx^{2}++3h^{2}x+h^{3}}{h}=3x^{2}+3hx+h^{2}$
(ii)

Hence

$f^{\prime}(x)=\lim_{h\rightarrow 0}\left(3x^{2}+3hx+h^{2}\right)=3x^{2}.$

Solution to Exercise 5:

	$\displaystyle\frac{d}{dx}\left(15x-\frac{2}{\sqrt[3]{x}}-\frac{x^{4}}{2}\right)$	$\displaystyle=\frac{d}{dx}\left(15x-2x^{\frac{1}{3}}-\frac{1}{2}x^{4}\right)$
		$\displaystyle=15-2\cdot\frac{1}{3}x^{-\frac{2}{3}}-\frac{1}{2}\cdot 4x^{3}=15-% \frac{2}{3}x^{-\frac{2}{3}}-2x^{3}.$