Calculatrice de dérivées partielles

Calculer les dérivées partielles étape par étape

Cette calculatrice en ligne calculera la dérivée partielle de la fonction, avec les étapes indiquées. Vous pouvez spécifier n'importe quel ordre d'intégration.

Solution

Your input: find $\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)$

First, find $\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)$

The derivative of a sum/difference is the sum/difference of derivatives:

${\color{red}{\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)}}={\color{red}{\left(\frac{\partial}{\partial x}\left(e^{x}\right) + \frac{\partial}{\partial x}\left(e^{y}\right)\right)}}$

The derivative of a constant is 0:

${\color{red}{\frac{\partial}{\partial x}\left(e^{y}\right)}} + \frac{\partial}{\partial x}\left(e^{x}\right)={\color{red}{\left(0\right)}} + \frac{\partial}{\partial x}\left(e^{x}\right)$

The derivative of an exponential is $\frac{\partial}{\partial x} \left(e^{x} \right)=e^{x}$ :

${\color{red}{\frac{\partial}{\partial x}\left(e^{x}\right)}}={\color{red}{e^{x}}}$

Thus, $\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)=e^{x}$

Next, $\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial x} \left(\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial x}\left(e^{x}\right)$

The derivative of an exponential is $\frac{\partial}{\partial x} \left(e^{x} \right)=e^{x}$ :

${\color{red}{\frac{\partial}{\partial x}\left(e^{x}\right)}}={\color{red}{e^{x}}}$

Thus, $\frac{\partial}{\partial x}\left(e^{x}\right)=e^{x}$

Next, $\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial y} \left(\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial y}\left(e^{x}\right)$

The derivative of a constant is 0:

${\color{red}{\frac{\partial}{\partial y}\left(e^{x}\right)}}={\color{red}{\left(0\right)}}$

Thus, $\frac{\partial}{\partial y}\left(e^{x}\right)=0$

Therefore, $\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=0$

Answer: $\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=0$

Calculatrice de dérivées partielles

Calculer les dérivées partielles étape par étape

Solution

First, find ∂∂x(ex+ey)\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)

Next, ∂2∂x2(ex+ey)=∂∂x(∂∂x(ex+ey))=∂∂x(ex)\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial x} \left(\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial x}\left(e^{x}\right)

Next, ∂3∂x2∂y(ex+ey)=∂∂y(∂2∂x2(ex+ey))=∂∂y(ex)\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial y} \left(\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial y}\left(e^{x}\right)

First, find $\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right)$

Next, $\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial x} \left(\frac{\partial}{\partial x}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial x}\left(e^{x}\right)$

Next, $\frac{\partial^{3}}{\partial x^{2} \partial y}\left(e^{x} + e^{y}\right)=\frac{\partial}{\partial y} \left(\frac{\partial^{2}}{\partial x^{2}}\left(e^{x} + e^{y}\right) \right)=\frac{\partial}{\partial y}\left(e^{x}\right)$