To address the partial derivatives for the wind chill function $C(T, w)$, let's break down the problem into a few key steps.

Partial Derivatives:

• $C_T(T, w)$ refers to the partial derivative of $C$ with respect to the temperature $T$ while keeping the wind speed $w$ constant.
• $C_w(T, w)$ refers to the partial derivative of $C$ with respect to the wind speed $w$ while keeping the temperature $T$ constant.

Estimating the Partial Derivatives Using the Difference Quotients:

For approximations, we use the following methods:

1. Forward Difference Quotient:
   $f_T(a, b) \approx \frac{f(a+h, b) - f(a, b)}{h}$ $f_w(a, b) \approx \frac{f(a, b+h) - f(a, b)}{h}$
2. Backward Difference Quotient:
   $f_T(a, b) \approx \frac{f(a, b) - f(a-h, b)}{h}$ $f_w(a, b) \approx \frac{f(a, b) - f(a, b-h)}{h}$
3. Symmetric Difference Quotient:
   $f_T(a, b) \approx \frac{f(a+h, b) - f(a-h, b)}{2h}$ $f_w(a, b) \approx \frac{f(a, b+h) - f(a, b-h)}{2h}$

Using the Formula for $C(T, w)$:

The given formula for $C(T, w)$ is: $C(T, w) \approx 35.74 + 0.6215T + w^{0.16} \left( -35.75 + 0.4275T \right)$ This formula will allow us to compute the analytical derivatives for $C_T(30, 15)$ and ( C_w(30,