Définitions et notations des dérivées partielles du second ordre
Des exemples avec des solutions détaillées sur la façon de calculer des dérivées partielles du second ordre sont présentés.
Pour une fonction de deux variables \( f(x, y) \), nous pouvons définir quatre dérivées partielles du second ordre ainsi que leurs notations standard :
\[ \begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial x} (f_x) = (f_x)_x = f_{xx} \\\\ \dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial y} (f_y) = (f_y)_y = f_{yy} \\\\ \dfrac{\partial^2 f}{\partial y \partial x} &= \dfrac{\partial}{\partial y} \left( \dfrac{\partial f}{\partial x} \right) = \dfrac{\partial}{\partial y} (f_x) = (f_x)_y = f_{xy} \\\\ \dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial y} \right) = \dfrac{\partial}{\partial x} (f_y) = (f_y)_x = f_{yx} \end{aligned} \]Exemples avec solutions détaillées sur les dérivées partielles du second ordre
Cliquez sur chaque exemple pour afficher sa solution détaillée étape par étape.
Exemple 1
Trouvez \( f_{xx} \) et \( f_{yy} \) étant donné que \( f(x, y) = \sin(xy) \).
Afficher la solution de l'exemple 1
Calcul de \( f_{xx} \) :
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}\sin(xy)\right) = \frac{\partial}{\partial x}(y \cos(xy)) = -y^2 \sin(xy) \]Calcul de \( f_{yy} \) :
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\sin(xy)\right) = \frac{\partial}{\partial y}(x \cos(xy)) = -x^2 \sin(xy) \]Exemple 2
Trouvez \( f_{xx} \), \( f_{yy} \), \( f_{xy} \) et \( f_{yx} \) étant donné que \( f(x, y) = x^3 + 2xy \).
Afficher la solution de l'exemple 2
Calcul de \( f_{xx} \) :
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(3x^2 + 2y) = 6x \]Calcul de \( f_{yy} \) :
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(2x) = 0 \]Calcul de \( f_{xy} \) :
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3 + 2xy)\right) = \frac{\partial}{\partial y}(3x^2 + 2y) = 2 \]Calcul de \( f_{yx} \) :
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3 + 2xy)\right) = \frac{\partial}{\partial x}(2x) = 2 \]Exemple 3
Trouvez \( f_{xx} \), \( f_{yy} \), \( f_{xy} \) et \( f_{yx} \) étant donné que \( f(x, y) = x^3y^4 + x^2y \).
Afficher la solution de l'exemple 3
Calcul de \( f_{xx} \) :
\[ f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(3x^2y^4 + 2xy) = 6xy^4 + 2y \]Calcul de \( f_{yy} \) :
\[ f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(4x^3y^3 + x^2) = 12x^3y^2 \]Calcul de \( f_{xy} \) :
\[ f_{xy} = \frac{\partial^2 f}{\partial y\partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial}{\partial y}\left(\frac{\partial}{\partial x}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial y}(3x^2y^4 + 2xy) = 12x^2y^3 + 2x \]Calcul de \( f_{yx} \) :
\[ f_{yx} = \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial x}\left(\frac{\partial}{\partial y}(x^3y^4 + x^2y)\right) = \frac{\partial}{\partial x}(4x^3y^3 + x^2) = 12x^2y^3 + 2x \]