Question

Another derivatives question, can I get help please?

Answer 1

$\bf \textit{de}\textit{finition of a derivative as a limit}\\\\ \lim\limits_{h\to 0}~\cfrac{f(x+h)-f(x)}{h}\\\\ -------------------------------\\\\ f(x)=5x^2-6x-3 \\\\\\ \lim\limits_{h\to 0}~\cfrac{[5(x+h)^2-6(x+h)-3]~~-~~(5x^2-6x-3)}{h} \\\\\\ \lim\limits_{h\to 0}~\cfrac{[5(x^2+2xh+h^2)-6(x+h)-3]~~-~~5x^2+6x+3}{h}$

$\bf \lim\limits_{h\to 0}~\cfrac{\underline{5x^2}+10xh+5h^2\underline{-6x}-6h\underline{-3}~~~\underline{-5x^2+6x+3}}{h} \\\\\\ \lim\limits_{h\to 0}~\cfrac{10xh+5h^2-6h}{h}\implies \lim\limits_{h\to 0}~\cfrac{\underline{h}(10x+5h-6)}{\underline{h}} \\\\\\ \lim\limits_{h\to 0}~10x+5(0)-6\implies \lim\limits_{h\to 0}~10x-6$