2024/G9/12

$\mathbf {Chapter3Section2}$
${\frac {d}{dx}}[c]=0$
${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]$
${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]$
${\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}$
${\frac {d}{dx}}[e^{x}]=e^{x}$
$\color {Blue}PowerRule$
${\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1$
$\color {Blue}ProductRule$
${\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}$
$\color {Blue}QuotientRule$
${\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}$
$\mathbf {Ex.1}$
$f(x)=x\cdot {e^{x}}$
$f^{\prime }(x)=1\cdot {e^{x}}+x\cdot {e^{x}}$
$\mathbf {Ex.2}$
$f(t)={\sqrt {t}}(a+bt)$
$f^{\prime }(t)={\frac {1}{2{\sqrt {t}}}}(a+bt)+t{\sqrt {t}}(b)$
$\mathbf {Ex.3}$
$if\,f(x)={\sqrt {x}}\cdot {g(x)}$
$g(4)=2$
$g^{\prime }(4)=3$ $f^{\prime }(x)=1{\sqrt {1}}{2{\sqrt {x}}}\cdot {g(x)}+{\sqrt {x}}\cdot {g^{\prime }(x)}$