Answer
(a) $f'(x) = 4x^3+2$
(b) By comparing the two graphs $f(x)$ and $f'(x)$, the answer to part (a) seems reasonable.
Work Step by Step
(a) $f(x) = x^4+2x$
We can find an expression for $f'(x)$:
$f'(x) = \lim\limits_{h \to 0}\frac{(x+h)^4+2(x+h)-(x^4+2x)}{h}$
$f'(x) = \lim\limits_{h \to 0}\frac{(x^4+4x^3h+6x^2h^2+4xh^3+h^4+2x+2h)-(x^4+2x)}{h}$
$f'(x) = \lim\limits_{h \to 0}\frac{(4x^3h+6x^2h^2+4xh^3+h^4+2h)}{h}$
$f'(x) = \lim\limits_{h \to 0}\frac{h(4x^3+6x^2h+4xh^2+h^3+2)}{h}$
$f'(x) = \lim\limits_{h \to 0}(4x^3+6x^2h+4xh^2+h^3+2)$
$f'(x) = 4x^3+0+0+0+2$
$f'(x) = 4x^3+2$
(b) By comparing the two graphs $f(x)$ and $f'(x)$, the answer to part (a) seems reasonable.
