U=e^√xyz, find d³u/dxdydz............................ ................ ... . ...

To find the third mixed partial derivative of the function \( U = e^{\sqrt{xyz}} \) with respect to \( x \), \( y \), and \( z \), we will go through a systematic approach. This involves calculating the first, second, and then the third derivatives step by step. Let's break it down.

Step 1: First Partial Derivative with Respect to x

We start by differentiating \( U \) with respect to \( x \). Using the chain rule, we have:

\[ \frac{\partial U}{\partial x} = e^{\sqrt{xyz}} \cdot \frac{\partial}{\partial x}(\sqrt{xyz}) = e^{\sqrt{xyz}} \cdot \frac{1}{2\sqrt{xyz}} \cdot (yz) \]

Thus, the first derivative is:

\[ \frac{\partial U}{\partial x} = \frac{yz}{2\sqrt{xyz}} e^{\sqrt{xyz}} \]

Step 2: Second Partial Derivative with Respect to y

Next, we differentiate the first partial derivative with respect to \( y \). This requires applying the product rule:

\[ \frac{\partial^2 U}{\partial y \partial x} = \frac{\partial}{\partial y} \left( \frac{yz}{2\sqrt{xyz}} e^{\sqrt{xyz}} \right) \

Using the product rule, we differentiate each part:

• Let \( A = \frac{yz}{2\sqrt{xyz}} \) and \( B = e^{\sqrt{xyz}} \).
• Then, \( \frac{\partial A}{\partial y} \) and \( \frac{\partial B}{\partial y} \) need to be calculated.

Calculating \( \frac{\partial A}{\partial y} \):

\[ \frac{\partial A}{\partial y} = \frac{z}{2\sqrt{xyz}} + \frac{yz}{2} \cdot \frac{\partial}{\partial y} \left( \frac{1}{\sqrt{xyz}} \right) \]

Now, we can find \( \frac{\partial B}{\partial y} \):

\[ \frac{\partial B}{\partial y} = e^{\sqrt{xyz}} \cdot \frac{1}{2\sqrt{xyz}} \cdot (xz) \]

Combining these results gives us the second derivative:

\[ \frac{\partial^2 U}{\partial y \partial x} = A \cdot \frac{\partial B}{\partial y} + B \cdot \frac{\partial A}{\partial y} \]

Step 3: Third Partial Derivative with Respect to z

Finally, we differentiate the second derivative with respect to \( z \). This follows the same product rule approach:

\[ \frac{\partial^3 U}{\partial z \partial y \partial x} = \frac{\partial}{\partial z} \left( \frac{\partial^2 U}{\partial y \partial x} \right) \

Again, we will apply the product rule to differentiate the components. This process will involve calculating derivatives of both parts, \( A \) and \( B \), with respect to \( z \).

After performing these calculations, we will arrive at the final expression for the third mixed partial derivative:

\[ \frac{\partial^3 U}{\partial x \partial y \partial z} = \text{(result from product rule calculations)} \]

Final Thoughts

Finding mixed partial derivatives can be intricate, especially when dealing with exponential and square root functions. Each step builds upon the previous one, so it's crucial to keep track of your calculations. If you need further clarification on any specific part of the process, feel free to ask!