To find the partial differential equation from \( P = e^q \), we need to differentiate this equation with respect to the variables involved.

Let \( P \) be a function of two independent variables, say \( x \) and \( y \), and \( q \) is a function of \( x \) and \( y \) (i.e., \( q = q(x, y) \)).

### Step 1: Differentiate with respect to \( x \)
Using the chain rule:
\[
\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(e^q) = e^q \cdot \frac{\partial q}{\partial x}
\]
Thus:
\[
P_x = e^q q_x
\]

### Step 2: Differentiate with respect to \( y \)
Similarly, differentiating with respect to \( y \):
\[
\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(e^q) = e^q \cdot \frac{\partial q}{\partial y}
\]
Thus:
\[
P_y = e^q q_y
\]

### Conclusion:
The partial differential equation that corresponds to \( P = e^q \) is expressed through the derivatives of \( P \) and \( q \). The system of PDEs we get is:

\[
P_x = P q_x
\]
and
\[
P_y = P q_y
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. What if \( P \) and \( q \) also depend on time?
2. How would you solve this system of PDEs?
3. Can we express \( q \) explicitly as a function of \( P \)?
4. What happens if \( P = e^{q(x)} + e^{q(y)} \)?
5. How can boundary conditions affect the solution to these PDEs?

**Tip:** Always check if the variables in the equation are interdependent before differentiating!

P = e^q find partial differential equation

This page provides a step-by-step guide to finding the partial differential equation for P = e^q using the chain rule. The problem explores partial derivatives and multivariable calculus, offering a detailed solution for students at the undergraduate level studying Calculus III or Differential Equations.

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