tangent line to 2x^3-5x at x=-1(b) 접선의 기울기 $m=\lim\limits_{h\to0}\dfrac{2h/(h+1)^2-0}{h}=\lim\limits_{h\to0}\dfrac{2}{(h+1)^2}=2$이므로 접선의 방정식은 $y-0=2(x-0)$, 즉 $y=2x$이다.
tangent line to 2x/(x+1)^2 at x=0
derivative of 1-x^3 at x=0(b) 접선의 방정식은 $y-1=0(x-0)$, 즉 $y=1$이다.
tangent line to 1-x^3 at x=0(c) 곡선 $y=1-x^3$과 접선 $y=1$을 동시에 그리는 Wolfram|Alpha 명령은 다음과 같다.
plot y=1-x^3 and y=1
derivative of sqrt(1+2x)(b) $\begin{aligned}[t] g'(x)&=\lim\limits_{h\to0}\dfrac{g(x+h)-g(x)}{h}=\lim\limits_{h\to0}\dfrac{\frac{4(x+h)}{(x+h)+1}-\frac{4x}{x+1}}{h}=\lim\limits_{h\to0}\dfrac{\frac{4h}{(x+h+1)(x+1)}}{h} \\ &=\lim\limits_{h\to0}\dfrac{4}{(x+h+1)(x+1)}=\dfrac{4}{(x+1)^2} \end{aligned}$
derivative of 4x/(x+1)