Question Number 12630 by sin (x) last updated on 27/Apr/17
$${f}\left({x}\right)=\left({x}^{} −\mathrm{3}\right)^{\mathrm{10}} \\ $$$$ \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left(\mathrm{2}+\mathrm{5}{h}\right)−{f}\left(\mathrm{2}+\mathrm{3}{h}\right)}{{h}}=? \\ $$
Answered by ajfour last updated on 27/Apr/17
$$=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{5}{hf}'\left(\mathrm{2}\right)−\mathrm{3}{hf}'\left(\mathrm{2}\right)}{{h}} \\ $$$$=\:\mathrm{2}{f}'\left(\mathrm{2}\right)\:=\mathrm{2}\left(\mathrm{10}\right)\left(\mathrm{2}−\mathrm{3}\right)^{\mathrm{9}} \:=\:−\mathrm{20}\:. \\ $$