Title: Calcul / Fonctions

Q: \(\cos\)
A: - \(\cos:\mathbb R\to[-1;1]\)
- \(\cos'=-\sin\)

Q: \(\sin\)
A: - \(\sin:\mathbb R\to[-1;1]\)
- \(\sin'=\cos\)

Q: \(\tan\)
A: - \(\tan:\mathbb R\backslash\left\{\frac{\pi}{2}+k\pi,k\in\mathbb Z\right\}\to\mathbb R\)
- \(\tan'=\frac{1}{\cos^2}=1+\tan^2\)

Q: \(\cot\)
A: - \(\cot:\mathbb R\backslash\pi\mathbb Z\to\mathbb R\)
- \(\cot'=-\frac{1}{\sin^2}=-(1+\cot^2)\)

Q: \(\arccos\)
A: - \(\arccos:[-1;1] \to [0;\pi]\)
- \(\arccos'=-\frac{1}{\sqrt{1-x^2}}\)

Q: \(\arcsin\)
A: - \(\arcsin:[-1;1]\to[-\frac{\pi}{2};\frac{\pi}{2}]\)
- \(\arcsin'=\frac{1}{\sqrt{1-x^2}}\)

Q: \(\arctan\)
A: - \(\arctan : \mathbb R\to\left]-\frac{\pi}{2};\frac{\pi}{2}\right[\)
- \(\arctan'=\frac{1}{1+x^2}\)

Q: \(\cosh\)
A: - \(\cosh:\mathbb R\to[1;+\infty[\)
- \(\cosh'=\sinh\)

Q: \( \sinh\)
A: - \(\sinh:\mathbb R\to\mathbb R\)
- \(\sinh'=\cosh\)

Q: \(\tanh\)
A: - \(\tanh:\mathbb R\to]-1;1[\)
- \(\tanh'=\frac{1}{\cosh^2}=1-\tanh^2\)

Q: \(\coth\)
A: - \(\coth:\mathbb R^*\to\mathbb R\backslash[-1;1]\)
- \(\coth'=-\frac{1}{\sinh^2}=1-\coth^2\)

Q: \(\text{argch}\)
A: - \(\text{argch}:[1:+\infty]\to\mathbb R^+\)
- \(\text{argch}'=\frac{1}{\sqrt{x^2-1}}\)
- \(\text{argch}(x)=\ln(x+\sqrt{x^2-1})\)

Q: \(\text{argsh}\)
A: - \(\text{argsh}:\mathbb R\to\mathbb R\)
- \(\text{argsh}'=\frac{1}{\sqrt{x^2+1}}\)
- \(\text{argsh}(x)=\ln(x+\sqrt{x^2+1})\)

Q: \(\text{argth}\)
A: - \(\text{argth}:]-1;1[\to\mathbb R\)
- \(\text{argth}'=\frac{1}{1-x^2}\)
- \(\text{argth}=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)\)