2024/G2/12

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{3.1}}
1. The derivative of a constant is 0 = 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}} [c] = 0 }
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^n] = nx^{n-1} }

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

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

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

6. 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}} [e^x] = e^x }


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{\color{Blue}{Examples}}}

1) 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}} [5 + \pi] = 0+0=0 }

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}} [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) Failed to parse (syntax error): {\displaystyle {\frac{d}{dx}} [sqrt(x)] = {\frac{d}{dx}} [x^{{\frac{1}{2}}] = {\frac{1}{2x}^{{\frac{1}{2}-1}} }

\sqrt{x}

${\frac {d}{dx}}[c]=0$
${\frac {d}{dx}}[c\cdot f(x)]=c\cdot {\frac {d}{dx}}[f(x)]$
${\frac {d}{dx}}[f(x)\pm g(x)]={\frac {d}{dx}}[f(x)]\pm {\frac {d}{dx}}[g(x)]$
${\frac {d}{dx}}[a^{x}]=\ln(a)a^{x}$
${\frac {d}{dx}}[e^{x}]=e^{x}$
$\color {Blue}Power\,Rule$
${\frac {d}{dx}}[x^{n}]=n\cdot x^{n}-1$
$\color {Red}Product\,Rule$
${\frac {d}{dx}}[f\cdot {g}]={\frac {d}{dx}}[f]\cdot {g}+{\frac {d}{dx}}[g]\cdot {f}$
$\color {Green}Quotient\,Rule$
${\frac {d}{dx}}[{\frac {f}{g}}]={\frac {{\frac {d}{dx}}[f]\cdot {g}-{\frac {d}{dx}}[g]\cdot {f}}{g^{2}}}$
$\mathbf {\color {Purple}{Examples}}$
$\mathbf {Ex.1}$
$if\,f(x)=x\cdot {e^{x}}$
$f^{\prime }(x)=1\cdot {e^{x}}+x\cdot {e^{x}}$
$\mathbf {Ex.2}$
$if\,f(t)={\sqrt {t}}(a+bt)$
$f^{\prime }(t)={\frac {1}{2{\sqrt {t}}}}(a+bt)+t{\sqrt {t}}(b)$
$\mathbf {Ex.3}$
$if\,f(x)={\sqrt {x}}\cdot {g(x)}$
$g(4)=2$
$g^{\prime }(4)=3$
$f^{\prime }(x)={\frac {1}{2{\sqrt {x}}}}\cdot {g(x)}+{\sqrt {x}}\cdot {g^{\prime }(x)}$
$\mathbf {Ex.4}$
$y={\frac {\color {Blue}{x^{2}+x-2}}{\color {Red}{x^{3}+6}}}$
${\frac {d}{dx}}=y^{\prime }={\frac {(2x+1)(x^{3}-6)-\color {Blue}{(x^{2}+x-2)}(3x^{2})}{\color {Red}{(x^{3}+6)^{2}}}}$
$={\frac {(2x^{4}+x^{4}+x^{3}+12x+6-[3x^{4}+3x^{2}-6x^{2}]}{(x^{3}+6)^{2}}}$
$={\frac {-x^{4}-2x^{3}+6x^{2}+12x+6}{(x^{3}+6)^{2}}}$
$\mathbf {Ex.5}$
$y={\frac {e^{x}}{1+x^{2}}}\,(1,{\frac {e}{2}})\,$
${\frac {d}{dx}}={\frac {e^{x}\cdot (1+x^{2})-e^{x}(2x)}{(1+x^{2})^{2}}}$
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}