2024/G2/12

$\mathbf {3.1}$
1. The derivative of a constant is 0 = ${\frac {d}{dx}}[c]=0$
2. ${\frac {d}{dx}}[x^{n}]=nx^{n-1}$

3. ${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]$

4. ${\frac {d}{dx}}[f(x)+g(x)]={\frac {d}{dx}}[f(x)]+{\frac {d}{dx}}[g(x)]$

5. ${\frac {d}{dx}}[a^{x}]=ln(a)\cdot a^{x}$

6. ${\frac {d}{dx}}[e^{x}]=e^{x}$


$\mathbf {\color {Blue}{Examples}}$

1) ${\frac {d}{dx}}[5+\pi ]=0+0=0$

2) ${\frac {d}{dx}}[3x]=3\cdot {\frac {d}{dx}}[x]=3(1)(x)^{1-1}=3\cdot x^{0}=3$

3) ${\frac {d}{dx}}[9x^{2}]=18x$

4) ${\frac {d}{dx}}[x^{3}+x^{2}+10]=3x^{2}+2x+0$
5) ${\frac {d}{dx}}[{\sqrt {(}}x)]={\frac {d}{dx}}[x^{\frac {1}{2}}]={\frac {1}{2x}}^{{\frac {1}{2}}-{\frac {2}{2}}}={\frac {1}{2x}}^{\frac {-1}{2}}={\frac {1}{2{\sqrt {(}}x)}}$
A1.

${\sqrt {(}}x)$

\sqrt{x}