Dérivées - Différentielles

Par définition \(\Delta f = f(x_0+\Delta x)-f(x_0)\) et \(df=f'(x_0)dx=\frac{2}{3}x_0^{-\frac{1}{3}\Delta x}\)

• si \(\Delta x = 0,1\)

  \(\Delta f = (1,1)^{\frac{2}{3}}-1=0,065\)

  \(df=(2/3).0,1=0,066 \Rightarrow \color{blue}D \# 10^{-3}~~\color{red}\textrm{(2 points)}\)

• si \(\Delta x = 0,01\)

  \(\Delta f=(1,01)^{\frac{2}{3}}-1=0,00665\)

  \(df=(2/3)0,01=0,00666\Rightarrow \color{blue}D \# 10^{-5}~~\color{red}\textrm{(2 points)}\)

• si \(\Delta x = 0,001\)

  \(\Delta f = (1,001)^{\frac{2}{3}}-1=0,0006665\)

  \(df=(2/3)0,001=0,0006666\Rightarrow \color{blue} D \# 10^{-7}~~\color{red}\textrm{(2 points)}\)

On remarque que \(D \rightarrow 0\) au fur et à mesure que \(\Delta x \rightarrow 0\) et \(\Delta f \cong df~~\color{red}\textrm{(2 points)}\)