Question:
Find the derivative of $x^{2}-2$ at $x=10$.
Solution:
Let $f(x)=x^{2}-2$. Accordingly,
$f^{\prime}(10)=\lim _{h \rightarrow 0} \frac{f(10+h)-f(10)}{h}$
$=\lim _{h \rightarrow 0} \frac{\left[(10+h)^{2}-2\right]-\left(10^{2}-2\right)}{h}$
$=\lim _{h \rightarrow 0} \frac{10^{2}+2 \cdot 10 \cdot h+h^{2}-2-10^{2}+2}{h}$
$=\lim _{h \rightarrow 0} \frac{20 h+h^{2}}{h}$
$=\lim _{h \rightarrow 0}(20+h)=(20+0)=20$
Thus, the derivative of $x^{2}-2$ at $x=10$ is 20 .