find dF, dG if F=x³y-4xy²+8y³ and G=8xy²z³-3x²yz

To find the differentials \( dF \) and \( dG \) for the functions \( F(x, y) = x^3y - 4xy^2 + 8y^3 \) and \( G(x, y, z) = 8xy^2z^3 - 3x^2yz \), we need to compute the partial derivatives of each function with respect to each variable and then use the differential formula:

\[
dF = \frac{\partial F}{\partial x} dx + \frac{\partial F}{\partial y} dy
\]

\[
dG = \frac{\partial G}{\partial x} dx + \frac{\partial G}{\partial y} dy + \frac{\partial G}{\partial z} dz
\]

### Finding \( dF \):
1. **Partial derivative of \( F \) with respect to \( x \):**
   \[
   \frac{\partial F}{\partial x} = \frac{\partial}{\partial x} \left( x^3y - 4xy^2 + 8y^3 \right)
   \]
   \[
   \frac{\partial F}{\partial x} = 3x^2y - 4y^2
   \]

2. **Partial derivative of \( F \) with respect to \( y \):**
   \[
   \frac{\partial F}{\partial y} = \frac{\partial}{\partial y} \left( x^3y - 4xy^2 + 8y^3 \right)
   \]
   \[
   \frac{\partial F}{\partial y} = x^3 - 8xy + 24y^2
   \]

So, the differential \( dF \) is:
\[
dF = (3x^2y - 4y^2)dx + (x^3 - 8xy + 24y^2)dy
\]

### Finding \( dG \):
1. **Partial derivative of \( G \) with respect to \( x \):**
   \[
   \frac{\partial G}{\partial x} = \frac{\partial}{\partial x} \left( 8xy^2z^3 - 3x^2yz \right)
   \]
   \[
   \frac{\partial G}{\partial x} = 8y^2z^3 - 6xyz
   \]

2. **Partial derivative of \( G \) with respect to \( y \):**
   \[
   \frac{\partial G}{\partial y} = \frac{\partial}{\partial y} \left( 8xy^2z^3 - 3x^2yz \right)
   \]
   \[
   \frac{\partial G}{\partial y} = 16xyz^3 - 3x^2z
   \]

3. **Partial derivative of \( G \) with respect to \( z \):**
   \[
   \frac{\partial G}{\partial z} = \frac{\partial}{\partial z} \left( 8xy^2z^3 - 3x^2yz \right)
   \]
   \[
   \frac{\partial G}{\partial z} = 24xy^2z^2 - 3x^2y
   \]

So, the differential \( dG \) is:
\[
dG = (8y^2z^3 - 6xyz)dx + (16xyz^3 - 3x^2z)dy + (24xy^2z^2 - 3x^2y)dz
\]

### Summary:
- \( dF = (3x^2y - 4y^2)dx + (x^3 - 8xy + 24y^2)dy \)
- \( dG = (8y^2z^3 - 6xyz)dx + (16xyz^3 - 3x^2z)dy + (24xy^2z^2 - 3x^2y)dz \)

Would you like to see more details on these computations or have any further questions?

### Relative Questions:
1. How do you interpret the differential \( dF \) geometrically?
2. What does the differential tell us about the change in a function?
3. How would the differentials change if we included higher-order terms?
4. Can you derive the total differential for a function of more than three variables?
5. How do differentials relate to gradients in vector calculus?

### Tip:
When computing partial derivatives, always keep in mind which variables are held constant to avoid mistakes in your calculations.

Learn how to compute the differentials dF and dG for functions F(x, y) = x^3y - 4xy^2 + 8y^3 and G(x, y, z) = 8xy^2z^3 - 3x^2yz. This detailed solution involves calculating partial derivatives and using differential formulas. Suitable for advanced undergraduate studies.

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