2024/G2/13

$\mathbf {3.2}$
${\frac {d}{dx}}[c]=0$
${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]$
${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]$
${\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}$
${\frac {d}{dx}}[e^{x}]=e^{x}$
$\color {Blue}Power\,Rule$
${\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1$
$\color {Red}Product\,Rule$
${\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}$
$\color {Green}Quotient\,Rule$
${\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}$
$\mathbf {\color {Purple}{Examples}}$
$\mathbf {Ex.1}$
$if\,f(x)=x\cdot {e^{x}}$
$f^{\prime }(x)=1\cdot {e^{x}}+x\cdot {e^{x}}$
$\mathbf {Ex.2}$
$if\,f(t)={\sqrt {t}}(a+bt)$
$f^{\prime }(t)={\frac {1}{2{\sqrt {t}}}}(a+bt)+t{\sqrt {t}}(b)$
$\mathbf {Ex.3}$
$if\,f(x)={\sqrt {x}}\cdot {g(x)}$
$g(4)=2$
$g^{\prime }(4)=3$
$f^{\prime }(x)={\frac {1}{2{\sqrt {x}}}}\cdot {g(x)}+{\sqrt {x}}\cdot {g^{\prime }(x)}$
$\mathbf {Ex.4}$
$y={\frac {\color {Blue}{x^{2}+x-2}}{\color {Red}{x^{3}+6}}}$
${\frac {d}{dx}}=y^{\prime }={\frac {(2x+1)(x^{3}-6)-\color {Blue}{(x^{2}+x-2)}(3x^{2})}{\color {Red}{(x^{3}+6)^{2}}}}$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{(2x^4+x^4+x^3+12x+6-[3x^4+3x^2-6x^2]}{(x^3+6)^2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{-x^4-2x^3+6x^2+12x+6}{(x^3+6)^2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{Ex.5}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y=\frac{e^x}{1+x^2}\,(1,\frac{e}{2})\,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}}=\frac{e^x\cdot(1+x^2)-e^x(2x)}{(1+x^2)^2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\frac{d}{dx}}|_{x=1}\frac{e(1+1)-e^\prime(2)}{(1+1)^2}=\frac{2e-2e}{2^2}=\frac{0}{4}=0}