Answer
$100 \pi $ cubic inches per inch
Work Step by Step
a) Now, $ fâ˛(x)=\lim_\limits{ h\to 0} \dfrac{f(x+h)-f(x)}{h}$
or, $=\lim_\limits{ h\to 0} \dfrac{\dfrac{4 \pi}{3}(x+h)^3-\dfrac{4 \pi}{3}x^3}{h}$
or, $=\lim_\limits{ h\to 0} \dfrac{\dfrac{4 \pi}{3}(x^3+3x^2h+3xh^2+h^3)-\dfrac{4 \pi}{3}x^3}{h}$
or, $=\lim_\limits{ h\to 0} \dfrac{(4 \pi x^2+4\pi xh+(4/3) \pi h^2}{h}$
or, $=4 \pi x^2$
and $ f(5)=4 \pi (5)^2=100 \pi $
