Problems from The Derivative function

Development:

According to the definition, $${\displaystyle f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}}$$ that is, $${\displaystyle f^{\prime }(x)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{(x+\mathrm{\Delta }x{)}^2+(x+\mathrm{\Delta }x)-(x^2+x)}{\mathrm{\Delta }x}=}$$ $$=\underset{\mathrm{\Delta }x\to 0}{lim}{\displaystyle \frac{x^2+2x\mathrm{\Delta }x+\mathrm{\Delta }{x}^2+x+\mathrm{\Delta }x-x^2-x}{\mathrm{\Delta }x}}=\underset{\mathrm{\Delta }x\to 0}{lim}(\mathrm{\Delta }x+2x+1)=2x+1$$

Solution:

$f^{\prime }(x)=2x+1$

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