If f is defined by


If $f$ is defined by $f(x)=x^{2}$, find $f(2)$.


Given: $f(x)=x^{2}$

We know a polynomial function is everywhere differentiable. Therefore $f(x)$ is differentiable at $x=2$.

$f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$

$\Rightarrow f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{(2+h)^{2}-2^{2}}{h}$

$\Rightarrow f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{\left(4+h^{2}+4 h\right)-4}{h}$

$\Rightarrow f^{\prime}(2)=\lim _{h \rightarrow 0} \frac{h(h+4)}{h}$

$\Rightarrow f^{\prime}(2)=4$

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