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  3. Calculatrices: Calcul différentiel et intégral III
  4. Calculatrice de calcul différentiel et intégral

Calculatrice de dérivées partielles

Calculez les dérivées partielles étape par étape

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

Enter a function:

Enter the order of integration:

Hint: type x^2,y to calculate `(partial^3 f)/(partial x^2 partial y)`, or enter x,y^2,x to find `(partial^4 f)/(partial x partial y^2 partial x)`.

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Solution

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

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

$${\color{red}{\frac{\partial}{\partial y}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)}}={\color{red}{\left(\frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 x^{2}\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right) + \frac{\partial}{\partial y}\left(2 x^{2} y\right)\right)}}$$

Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=2 x^{2}$$$ and $$$f=y$$$:

$${\color{red}{\frac{\partial}{\partial y}\left(2 x^{2} y\right)}} + \frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 x^{2}\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right)={\color{red}{2 x^{2} \frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 x^{2}\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right)$$

Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=1$$$, in other words $$$\frac{\partial}{\partial y} \left(y \right)=1$$$:

$$2 x^{2} {\color{red}{\frac{\partial}{\partial y}\left(y\right)}} + \frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 x^{2}\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right)=2 x^{2} {\color{red}{1}} + \frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 x^{2}\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right)$$

The derivative of a constant is 0:

$$2 x^{2} - {\color{red}{\frac{\partial}{\partial y}\left(2 x^{2}\right)}} + \frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right)=2 x^{2} - {\color{red}{\left(0\right)}} + \frac{\partial}{\partial y}\left(2\right) - \frac{\partial}{\partial y}\left(2 y^{2}\right) + \frac{\partial}{\partial y}\left(y^{3}\right)$$

Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=2$$$ and $$$f=y^{2}$$$:

$$2 x^{2} - {\color{red}{\frac{\partial}{\partial y}\left(2 y^{2}\right)}} + \frac{\partial}{\partial y}\left(2\right) + \frac{\partial}{\partial y}\left(y^{3}\right)=2 x^{2} - {\color{red}{\left(2 \frac{\partial}{\partial y}\left(y^{2}\right)\right)}} + \frac{\partial}{\partial y}\left(2\right) + \frac{\partial}{\partial y}\left(y^{3}\right)$$

Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=2$$$:

$$2 x^{2} - 2 {\color{red}{\frac{\partial}{\partial y}\left(y^{2}\right)}} + \frac{\partial}{\partial y}\left(2\right) + \frac{\partial}{\partial y}\left(y^{3}\right)=2 x^{2} - 2 {\color{red}{\left(2 y^{-1 + 2}\right)}} + \frac{\partial}{\partial y}\left(2\right) + \frac{\partial}{\partial y}\left(y^{3}\right)=2 x^{2} - 4 y + \frac{\partial}{\partial y}\left(2\right) + \frac{\partial}{\partial y}\left(y^{3}\right)$$

The derivative of a constant is 0:

$$2 x^{2} - 4 y + {\color{red}{\frac{\partial}{\partial y}\left(2\right)}} + \frac{\partial}{\partial y}\left(y^{3}\right)=2 x^{2} - 4 y + {\color{red}{\left(0\right)}} + \frac{\partial}{\partial y}\left(y^{3}\right)$$

Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=3$$$:

$$2 x^{2} - 4 y + {\color{red}{\frac{\partial}{\partial y}\left(y^{3}\right)}}=2 x^{2} - 4 y + {\color{red}{\left(3 y^{-1 + 3}\right)}}=2 x^{2} + 3 y^{2} - 4 y$$

Thus, $$$\frac{\partial}{\partial y}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)=2 x^{2} + 3 y^{2} - 4 y$$$

Answer: $$$\frac{\partial}{\partial y}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)=2 x^{2} + 3 y^{2} - 4 y$$$


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