Calcolatrice delle derivate parziali

Your input: find $\frac{\partial^{2}}{\partial x \partial y}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)$

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

$${\color{red}{\frac{\partial}{\partial x}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)}}={\color{red}{\left(- \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right) + \frac{\partial}{\partial x}\left(4 x y^{2}\right)\right)}}$$

Apply the constant multiple rule $\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$ with $c=4 y^{2}$ and $f=x$:

$${\color{red}{\frac{\partial}{\partial x}\left(4 x y^{2}\right)}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)={\color{red}{4 y^{2} \frac{\partial}{\partial x}\left(x\right)}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$

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

$$4 y^{2} {\color{red}{\frac{\partial}{\partial x}\left(x\right)}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)=4 y^{2} {\color{red}{1}} - \frac{\partial}{\partial x}\left(10\right) + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$

The derivative of a constant is 0:

$$4 y^{2} - {\color{red}{\frac{\partial}{\partial x}\left(10\right)}} + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)=4 y^{2} - {\color{red}{\left(0\right)}} + \frac{\partial}{\partial x}\left(x^{3}\right) + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$

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

$$4 y^{2} + {\color{red}{\frac{\partial}{\partial x}\left(x^{3}\right)}} + \frac{\partial}{\partial x}\left(5 y^{3}\right)=4 y^{2} + {\color{red}{\left(3 x^{-1 + 3}\right)}} + \frac{\partial}{\partial x}\left(5 y^{3}\right)=3 x^{2} + 4 y^{2} + \frac{\partial}{\partial x}\left(5 y^{3}\right)$$

The derivative of a constant is 0:

$$3 x^{2} + 4 y^{2} + {\color{red}{\frac{\partial}{\partial x}\left(5 y^{3}\right)}}=3 x^{2} + 4 y^{2} + {\color{red}{\left(0\right)}}$$

Thus, $\frac{\partial}{\partial x}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=3 x^{2} + 4 y^{2}$

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

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

Apply the constant multiple rule $\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$ with $c=4$ and $f=y^{2}$:

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

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

$$4 {\color{red}{\frac{\partial}{\partial y}\left(y^{2}\right)}} + \frac{\partial}{\partial y}\left(3 x^{2}\right)=4 {\color{red}{\left(2 y^{-1 + 2}\right)}} + \frac{\partial}{\partial y}\left(3 x^{2}\right)=8 y + \frac{\partial}{\partial y}\left(3 x^{2}\right)$$

The derivative of a constant is 0:

$$8 y + {\color{red}{\frac{\partial}{\partial y}\left(3 x^{2}\right)}}=8 y + {\color{red}{\left(0\right)}}$$

Thus, $\frac{\partial}{\partial y}\left(3 x^{2} + 4 y^{2}\right)=8 y$

Therefore, $\frac{\partial^{2}}{\partial x \partial y}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=8 y$

Answer: $\frac{\partial^{2}}{\partial x \partial y}\left(x^{3} + 4 x y^{2} + 5 y^{3} - 10\right)=8 y$