Problems from Derivative at a point

Development:

Using the definition $${\displaystyle f^{\prime }(a)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(a+\mathrm{\Delta }x)-f(a)}{\mathrm{\Delta }x}}$$ At the point $x=1$

$${\displaystyle f^{\prime }(1)=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{f(1+\mathrm{\Delta }x)-f(1)}{\mathrm{\Delta }x}=\underset{\mathrm{\Delta }x\to 0}{lim}\frac{(1+\mathrm{\Delta }x{)}^2+(1+\mathrm{\Delta }x)-(1^2+1)}{\mathrm{\Delta }x}=}$$ $$=\underset{\mathrm{\Delta }x\to 0}{lim}{\displaystyle \frac{\mathrm{\Delta }{x}^2+3\mathrm{\Delta }x}{\mathrm{\Delta }x}}=\underset{\mathrm{\Delta }x\to 0}{lim}(\mathrm{\Delta }x+3)=3$$

Solution:

$f^{\prime }(1)=3$

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