https://file.aiquickdraw.com/mathbot/file/1727102049191igy4j9fu.jpg

The text in the image is written in Indonesian. It seems to be asking for the proof of a partial differential equation.

The relevant part from the image reads:

**6) Jika \( w = f(ax + by) \), tunjukkan bahwa:**
\[
b \frac{\partial w}{\partial x} - a \frac{\partial w}{\partial y} = 0
\]

Translation:

**6) If \( w = f(ax + by) \), show that:**
\[
b \frac{\partial w}{\partial x} - a \frac{\partial w}{\partial y} = 0
\]

### Solution:

Let \( u = ax + by \). Then \( w = f(u) \).

#### Step 1: Compute \( \frac{\partial w}{\partial x} \)
Since \( w = f(u) \) and \( u = ax + by \), we apply the chain rule:
\[
\frac{\partial w}{\partial x} = \frac{df}{du} \cdot \frac{\partial u}{\partial x}
\]
We have:
\[
\frac{\partial u}{\partial x} = a
\]
Thus:
\[
\frac{\partial w}{\partial x} = a \frac{df}{du}
\]

#### Step 2: Compute \( \frac{\partial w}{\partial y} \)
Similarly, applying the chain rule:
\[
\frac{\partial w}{\partial y} = \frac{df}{du} \cdot \frac{\partial u}{\partial y}
\]
We have:
\[
\frac{\partial u}{\partial y} = b
\]
Thus:
\[
\frac{\partial w}{\partial y} = b \frac{df}{du}
\]

#### Step 3: Substitute into the original expression
We need to show:
\[
b \frac{\partial w}{\partial x} - a \frac{\partial w}{\partial y} = 0
\]
Substituting the values:
\[
b \cdot a \frac{df}{du} - a \cdot b \frac{df}{du} = 0
\]
This simplifies to:
\[
ab \frac{df}{du} - ab \frac{df}{du} = 0
\]
Thus, the equation is satisfied, and the statement is proven.

---

Would you like further details or clarifications?

Here are some related questions to explore further:
1. What is the significance of the chain rule in multivariable calculus?
2. Can you generalize this proof for a more complex function?
3. How does the choice of \( w = f(ax + by) \) affect the form of partial derivatives?
4. What happens if \( w \) is not a function of a linear combination of \( x \) and \( y \)?
5. How would this work if you used a different function for \( w \), such as trigonometric or exponential functions?

**Tip:** In problems like this, look for symmetry in expressions when applying partial derivatives to simplify the proof!

Jika w = f(ax + by), tunjukkan bahwa: b (∂w/∂x) - a (∂w/∂y) = 0

This problem demonstrates the application of partial derivatives and the chain rule in multivariable calculus. The task is to prove that b (∂w/∂x) - a (∂w/∂y) = 0, where w is a function of the linear combination ax + by. Step-by-step solution provided with detailed explanation.

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