Taula derivades-batx

Petites taules per fer derivades suficients de primer de batxillerat en general. S'han afegit algunes curiositats per qui vulgui saber més.

Taula dels diferents símbols i notacions utilitzats per a funcions, derivades, constants i constants conegudes.

Funcions	Derivades	Constant	Constants conegudes
${\begin{array}{c}f(x),g(x),h(x),\dots \\f,g,h,\dots \\u,v,w\\x,y,z\end{array}}$	${\begin{array}{rcl}f(x)&\to &f'(x)\\f&\to &Df\\f&\to &{\frac {df}{dx}}\\y&\to &y'\\u&\to &u'\\v&\to &v'\end{array}}$	${\begin{array}{c}k\\a,b,c,\dots \in \mathbb {R} \\m,n,\dots \in \mathbb {N} \\r,s,\dots \end{array}}$	${\begin{array}{l}\pi =3,141\,592\,653\,589\,793\dots \\e=2,718\,281\,828\,459\,045\dots \\\phi =1,618\,033\,988\,749\,894\dots \\{\sqrt {2}}=1,414\,213\,562\,373\,095\dots \end{array}}$

N°	Funció	Funció derivada	Exemples
1	$f(x)=k$	$f'(x)=0$	${\begin{array}{lcl}f(x)=4&\to &f'(x)=0\\f(x)=2\pi +e+{\sqrt {2}}&\to &f'(x)=0\\f(x)=Sin(2\pi +1)&\to &f'(x)=0\end{array}}$
2	$f(x)=x^{n}$	$f'(x)=n\cdot x^{n-1}$ on $n\neq 0$	${\begin{array}{lcl}f(x)=x^{-3}&\to &f'(x)=-3x^{-4}\\f(x)=x^{1}&\to &f'(x)=1x^{0}=1\\f(x)=x^{2}&\to &f'(x)=2x\\f(x)=x^{5}&\to &f'(x)=5x^{4}\end{array}}$
2.1	$f(x)=x$	$f'(x)=1$
3	$f(x)={\sqrt[{n}]{x}}$	$f'(x)={\frac {1}{n{\sqrt[{n}]{x^{n-1}}}}}$	${\begin{array}{lcl}f(x)={\sqrt[{2}]{x}}&\to &f'(x)={\frac {1}{2{\sqrt[{2}]{x}}}}\\f(x)={\sqrt[{3}]{x}}&\to &f'(x)={\frac {1}{3{\sqrt[{3}]{x^{2}}}}}\\f(x)={\sqrt[{7}]{x}}&\to &f'(x)={\frac {1}{7{\sqrt[{7}]{x^{6}}}}}\end{array}}$
3.1	$f(x)={\sqrt {x}}$	$f'(x)={\frac {1}{2{\sqrt {x}}}}$
4	$f(x)=a^{x}$	$f'(x)=a^{x}\cdot \ln a$	${\begin{array}{lcl}f(x)=4^{x}&\to &f'(x)=4^{x}\cdot \ln 4\\f(x)=3\cdot e^{x}-5x+4&\to &f'(x)=3e^{x}-5\end{array}}$
4.1	$f(x)=e^{x}$	$f'(x)=e^{x}$
5	$f(x)=\log _{a}x$	$f'(x)={\frac {1}{x\cdot \ln a}}$	${\begin{array}{lcl}f(x)=\log _{2}x&\to &f'(x)={\frac {1}{x\cdot \ln 2}}\\f(x)=x\ln x&\to &f'(x)=1\cdot \ln x+x{\frac {1}{x}}\end{array}}$
5.1	$f(x)=\ln x$	$f'(x)={\frac {1}{x}}$
6	$f(x)=\sin x$	$f'(x)=\cos x$	${\begin{array}{lcl}f(x)=5\sin x&\to &f'(x)=6\cos x\\f(x)=2\cos x-3\ln x&\to &f'(x)=-2\sin x-{\frac {3}{x}}\\f(x)={\frac {\tan x}{3}}-{\frac {2\cos x}{5}}&\to &f'(x)={\frac {1}{3\cos ^{2}x}}+{\frac {2\sin x}{5}}\end{array}}$
7	$f(x)=\cos x$	$f'(x)=-\sin x$
8	$f(x)=\tan x$	$f'(x)={\frac {1}{\cos ^{2}x}}$
9	$f(x)=\arcsin x$	$f'(x)={\frac {1}{\sqrt {1-x^{2}}}}$	${\begin{array}{lcl}f(x)=2\arcsin x&\to &f'(x)={\frac {2}{\sqrt {1-x^{2}}}}\\f(x)=x\cdot \arctan x&\to &f'(x)=\arctan x+{\frac {x}{1+x^{2}}}\end{array}}$
10	$f(x)=\arccos x$	$f'(x)=-{\frac {1}{\sqrt {1-x^{2}}}}$
11	$f(x)=\arctan x$	$f'(x)={\frac {1}{1+x^{2}}}$

N°	Operació	Funció derivada	Exemples
1	$u+v$	$(u+v)'=u'+v'$	${\begin{array}{lcl}f(x)=x^{4}+6&\to &f'(x)=4\cdot x^{3}+0=4x^{3}\\f(x)=\pi +e\cdot 3+{\sqrt {x}}&\to &f'(x)=0+0+{\frac {1}{2\cdot {\sqrt {x}}}}={\frac {1}{2{\sqrt {x}}}}\end{array}}$
2	$u\cdot v$	$(u\cdot v)'=u'\cdot v+u\cdot v'$	${\begin{array}{lcl}f(x)=x^{2}\cdot {\sqrt {x}}&\to &f'(x)=2x\cdot {\sqrt {x}}+x^{2}\cdot {\frac {1}{2{\sqrt {x}}}}=2x{\sqrt {x}}+x{\frac {\sqrt {x}}{2}}={\frac {5}{2}}x{\sqrt {x}}\\f(x)=5x^{2}+3x&\to &f'(x)=5(x^{2})'+3(x)'=5\cdot 2x+3\cdot 1=10x+3\end{array}}$
2.1	$k\cdot v$	$(k\cdot v)'=k\cdot v'$
3	${\frac {u}{v}}$	$\left({\frac {u}{v}}\right)'={\frac {u'\cdot v-u\cdot v'}{v^{2}}}$	${\begin{array}{lcl}f(x)={\frac {x^{2}}{\sqrt {x}}}&\to &f'(x)={\frac {2x\cdot {\sqrt {x}}-x^{2}\cdot {\frac {1}{2{\sqrt {x}}}}}{\left({\sqrt {x}}\right)^{2}}}={\frac {2x{\sqrt {x}}-x\cdot {\frac {1}{2}}{\sqrt {x}}}{x}}={\frac {3}{2}}{\sqrt {x}}\\f(x)={\frac {x}{3}}-{\frac {3x^{2}}{5}}&\to &f'(x)={\frac {1}{3}}-{\frac {6x}{5}}\\f(x)={\frac {1}{x}}&\to &f'(x)=-{\frac {1}{x^{2}}}\\f(x)={\frac {5}{x}}-{\frac {3}{x^{2}}}&\to &f'(x)=-{\frac {5}{x^{2}}}-\left(-{\frac {3\cdot 2x}{\left(x^{2}\right)^{2}}}\right)=-{\frac {5}{x^{2}}}+{\frac {6}{x^{3}}}\end{array}}$
3.1	${\frac {u}{k}}$	$\left({\frac {u}{k}}\right)'={\frac {u'}{k}}$
3.2	${\frac {k}{v}}$	$\left({\frac {k}{v}}\right)'={\frac {-k\cdot v'}{v^{2}}}$
4	$f\left(g(x)\right)$	$\left(f\left(g(x)\right)\right)'=f'(g(x))\cdot g'(x)$	Regla de la cadena $f(x)=\cos x^{2}$ $\to$ $f'(x)=-\sin x^{2}\cdot 2x$
4.1	$f\left(g(h(x))\right)$	$\left(f\left(g(h(x))\right)\right)'=f'(g(h(x)))\cdot g'(h(x))\cdot h'(x)$