Problems from The Derivative function

Given the function $$f(x)=x^2+x$$, compute the function derived from its definition.

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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:

$$f'(x)=2x+1$$

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