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The problem asks to estimate the value of the partial derivative \(\frac{\partial f}{\partial x}\) at the point \( (15, 6) \) using the table. To do this, we need to calculate the rate of change of \( f(x, y) \) with respect to \( x \) while holding \( y \) constant (specifically at \( y = 6 \)).

From the table, we look at the values of \( f(x, 6) \) for different \( x \):

- For \( x = 10 \), \( f(10, 6) = 133 \)
- For \( x = 15 \), \( f(15, 6) = 144 \)
- For \( x = 20 \), \( f(20, 6) = 153 \)

We estimate \(\frac{\partial f}{\partial x}(15, 6)\) by approximating it as a finite difference. Using central differences, we have:

\[
\frac{\partial f}{\partial x}(15, 6) \approx \frac{f(20, 6) - f(10, 6)}{20 - 10} = \frac{153 - 133}{10} = \frac{20}{10} = 2
\]

So, the estimated value of \(\frac{\partial f}{\partial x}(15, 6)\) is approximately \( 2.0 \).

Would you like further details or have any other questions?

Here are some related questions:
1. How do you interpret partial derivatives in the context of this table?
2. Why is central difference used for approximating derivatives?
3. What does \(\frac{\partial f}{\partial y}\) represent in this context?
4. How would the approximation change if a larger or smaller step were used?
5. What would happen if we used forward or backward difference instead?

**Tip**: When approximating derivatives from tables, always choose points as close as possible to the target value to minimize errors.

Utilize a tabela acima para estimar o valor de ∂f/∂x(15, 6). Escreva a resposta numérica com 1 dígito decimal.

Learn how to estimate the partial derivative of f with respect to x at the point (15, 6) using the table of values and central difference approximation. Understand the key concepts of multivariable calculus and apply finite difference methods for accurate estimation. Suitable for university-level students studying calculus.

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