A list of things you can't do...

$\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}$

Example:$\sqrt{4+16}=2\sqrt{5}\neq6=2+4=\sqrt{4}+\sqrt{16}$

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$\frac{a+b}{a+c} \neq \frac{1+b}{1+c}$

Example:$\frac{4+8}{4+2}=\frac{12}{6}=2 \neq 3 = \frac{9}{3}=\frac{1+8}{1+2}$

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$(a+b)^2 \neq a^2 + b^2$

Example:$(2+3)^2 = 25 \neq 13 = 2^2 + 3^2$

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$\frac{0}{0} \neq 0$

$\frac{0}{0}$ is undefined!

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$\frac{a}{b} \neq \frac{\log (a)}{\log (b)}$

Example:$\frac{100}{10}=10 \neq 2=\frac{\log (100)}{\log (10)}$

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$\frac{a}{b} \neq \frac{\sqrt{a}}{\sqrt{b}}$

Example:$\frac{4}{9} \neq \frac{2}{3} = \frac{\sqrt{4}}{\sqrt{9}}$

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$\frac{d}{dx}\frac{f(x)}{g(x)}\neq\frac{\frac{d}{dx}f(x)}{\frac{d}{dx}g(x)}$

Example:$\frac{d}{dx}\frac{x^3}{x^2}=\frac{d}{dx}x=1\neq\frac{3x}{2}=\frac{3x^2}{2x}=\frac{\frac{d}{dx}x^3}{\frac{d}{dx}x^2}$

In general, use the quotient rule:
$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f^{\prime}g-fg^{\prime}}{g^2}$