Given the function $$f(x)=x^2+x$$, compute the derivative in the point $$x=1$$.
See development and solution
Development:
Using the definition $$$\displaystyle f'(a)=\lim_{\Delta x \to 0}\frac{f(a+\Delta x)-f(a)}{\Delta x}$$$ At the point $$x=1$$
$$$\displaystyle f'(1)=\lim_{\Delta x \to 0}\frac{f(1+\Delta x)-f(1)}{\Delta x}=\lim_{\Delta x \to 0}\frac{(1+\Delta x)^2+(1+\Delta x)-(1^2+1)}{\Delta x}=$$$ $$$=\lim_{\Delta x \to 0}\dfrac{\Delta x^2+3\Delta x}{\Delta x}=\lim_{\Delta x \to 0}(\Delta x+3)=3 $$$
Solution:
$$f'(1)=3$$