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偏導數計算器
逐步計算偏導數
此線上計算器可計算函數的偏導數，並顯示步驟。您可以指定任意的積分順序。
Solution
Your input: find $$$\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(3 x + 4 y\right)$$$
First, find $$$\frac{\partial}{\partial l}\left(3 x + 4 y\right)$$$
The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial l}\left(3 x + 4 y\right)}}={\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial l}\left(3 x + 4 y\right)=0$$$
Next, $$$\frac{\partial^{2}}{\partial l \partial a}\left(3 x + 4 y\right)=\frac{\partial}{\partial a} \left(\frac{\partial}{\partial l}\left(3 x + 4 y\right) \right)=\frac{\partial}{\partial a}\left(0\right)$$$
The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial a}\left(0\right)}}={\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial a}\left(0\right)=0$$$
Next, $$$\frac{\partial^{3}}{\partial l \partial a \partial m}\left(3 x + 4 y\right)=\frac{\partial}{\partial m} \left(\frac{\partial^{2}}{\partial l \partial a}\left(3 x + 4 y\right) \right)=\frac{\partial}{\partial m}\left(0\right)$$$
The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial m}\left(0\right)}}={\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial m}\left(0\right)=0$$$
Next, $$$\frac{\partial^{4}}{\partial l \partial a \partial m \partial d}\left(3 x + 4 y\right)=\frac{\partial}{\partial d} \left(\frac{\partial^{3}}{\partial l \partial a \partial m}\left(3 x + 4 y\right) \right)=\frac{\partial}{\partial d}\left(0\right)$$$
The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial d}\left(0\right)}}={\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial d}\left(0\right)=0$$$
Next, $$$\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(3 x + 4 y\right)=\frac{\partial}{\partial a} \left(\frac{\partial^{4}}{\partial l \partial a \partial m \partial d}\left(3 x + 4 y\right) \right)=\frac{\partial}{\partial a}\left(0\right)$$$
The derivative of a constant is 0:
$${\color{red}{\frac{\partial}{\partial a}\left(0\right)}}={\color{red}{\left(0\right)}}$$Thus, $$$\frac{\partial}{\partial a}\left(0\right)=0$$$
Therefore, $$$\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(3 x + 4 y\right)=0$$$
Answer: $$$\frac{\partial^{5}}{\partial l \partial a \partial m \partial d \partial a}\left(3 x + 4 y\right)=0$$$