Given the function $$f(x)=x^2+x$$, compute the function derived from its definition.
See development and solution
Development:
According to the definition, $$$\displaystyle f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$$ that is, $$$\displaystyle f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\lim_{\Delta x \to 0}\frac{(x+\Delta x)^2+(x+\Delta x)-(x^2+x)}{\Delta x}=$$$ $$$=\lim_{\Delta x \to 0}\dfrac{x^2+2x\Delta x+\Delta x^2+x+\Delta x-x^2-x}{\Delta x}=\lim_{\Delta x \to 0}(\Delta x+2x+1)=2x+1$$$
Solution:
$$f'(x)=2x+1$$