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
$\begin{array}{c}f(x),g(x),h(x),\dots \\ f,g,h,\dots \\ u,v,w\\ x,y,z\end{array}$	$\begin{array}{ccc}f(x)& \to & f^{\prime }(x)\\ f& \to & Df\\ f& \to & \frac{df}{dx}\\ y& \to & y^{\prime }\\ u& \to & u^{\prime }\\ v& \to & v^{\prime }\end{array}$	$\begin{array}{c}k\\ a,b,c,\dots \in ℝ\\ m,n,\dots \in \mathbb{N}\\ r,s,\dots \end{array}$	$\begin{array}{c}\pi =3,141592653589793\dots \\ e=2,718281828459045\dots \\ \varphi =1,618033988749894\dots \\ \sqrt{2}=1,414213562373095\dots \end{array}$

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

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