Kalkulator pochodnej częściowej

Obliczanie pochodnych cząstkowych krok po kroku

Ten kalkulator online obliczy pochodną cząstkową funkcji, z pokazanymi krokami. Można określić dowolną kolejność całkowania.

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

Kalkulator pochodnej częściowej

Obliczanie pochodnych cząstkowych krok po kroku

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