Derivative (Examples)

Example 1

$f\text{'}(x)=\backslash lim\_\{h\backslash rightarrow\; 0\}\; \backslash frac\{f(x+h)-f(x)\}\{h\}\; =\; \backslash lim\_\{h\backslash rightarrow\; 0\}\; \backslash frac\{5-5\}\{h\}\; =\; 0$

Example 2

$f\text{'}(4)\; =\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{f(4+h)-f(4)\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{2(4+h)-3-(2\backslash cdot\; 4-3)\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{8+2h-3-8+3\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{2h\}\{h\}$

$=\; 2$

Example 3

$f\text{'}(x)\; =\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{f(x+h)-f(x)\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{(x+h)^2\; -\; x^2\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{x^2\; +\; 2xh\; +\; h^2\; -\; x^2\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{2xh\; +\; h^2\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}(2x\; +\; h)$

$=\; 2x$

Example 4

$f\text{'}(x)\; =\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{f(x+h)-f(x)\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{\backslash sqrt\{x+h\}\; -\; \backslash sqrt\{x\}\}\{h\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{(\backslash sqrt\{x+h\}\; -\; \backslash sqrt\{x\})(\backslash sqrt\{x+h\}\; +\; \backslash sqrt\{x\})\}\{h(\backslash sqrt\{x+h\}\; +\; \backslash sqrt\{x\})\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{x+h\; -\; x\}\{h(\backslash sqrt\{x+h\}\; +\; \backslash sqrt\{x\})\}$

$=\; \backslash lim\_\{h\backslash rightarrow\; 0\}\backslash frac\{1\}\{\backslash sqrt\{x+h\}\; +\; \backslash sqrt\{x\}\}$

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