Answer
$$ f(x) =x^{3}+5 x+4 $$
$ \Rightarrow $
$$\begin{aligned}
f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
&=\lim _{h \rightarrow 0} \frac{(x+h)^{3}+5(x+h)+4-\left(x^{3}+5 x+4\right)}{h} \\
&=\lim _{h \rightarrow 0} \frac{3 x^{2} h+3 x h^{2}+h^{3}+5 h}{h}\\
&=\lim _{h \rightarrow 0}\left(3 x^{2}+3 x h+h^{2}+5\right)\\
&=3 x^{2}+5
\end{aligned}$$
Work Step by Step
$$ f(x) =x^{3}+5 x+4 $$
$ \Rightarrow $
$$\begin{aligned}
f^{\prime}(x) &=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\
&=\lim _{h \rightarrow 0} \frac{(x+h)^{3}+5(x+h)+4-\left(x^{3}+5 x+4\right)}{h} \\
&=\lim _{h \rightarrow 0} \frac{3 x^{2} h+3 x h^{2}+h^{3}+5 h}{h}\\
&=\lim _{h \rightarrow 0}\left(3 x^{2}+3 x h+h^{2}+5\right)\\
&=3 x^{2}+5
\end{aligned}$$
