점 \(x=1\)에서 다음 함수의 미분계수를 구하라.
$$f(x)=x^2$$
풀이
\(\displaystyle f'(1)=\lim_{\Delta x\to0}\frac{f(1+\Delta x)-f(1)}{\Delta x}\)
\(\displaystyle=\lim_{\Delta x\to0}\frac{(1+\Delta x)^2-1}{\Delta x}=\lim_{\Delta x\to0}\frac{1+2\Delta x +\Delta x^2-1}{\Delta x}\)
\(\displaystyle=\lim_{\Delta x\to0}\frac{2\Delta x +\Delta x^2}{\Delta x}=\lim_{\Delta x\to0}\frac{\Delta x(2 +\Delta x)}{\Delta x}\)
\(\displaystyle=\lim_{\Delta x\to0}(2 +\Delta x)=2+0=2\)