In Exercises 3–10, use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables. 3. ∂f ∂s , ∂f ∂r ; f (x, y, z) = xy + z2, x = s2, y = 2rs, z = r2 4. ∂f ∂r , ∂f ∂t ; f (x, y, z) = xy + z2, x = r + s − 2t, y = 3rt, z = s2

I'll use subscript notation for brevity, i.e.

3.

By the chain rule,

4.

By the chain rule,

"it's not the heat, it's the humidity".   that's a partly valid phrase you may have heard in the summer, but it's actually both.   the heat index, also known as the apparent temperature, is what the temperature feels like to the human body when relative humidity is combined with the air temperature.   this has important considerations for the human body's comfort.   when the body gets too hot, it begins to perspire or sweat to cool itself off.   if the perspiration is not able to evaporate, the body cannot regulate its temperature.   evaporation is a cooling process.   when perspiration is evaporated off the body, it effectively reduces the body's temperature.   when the atmospheric moisture content (i.e. relative humidity) is high, the rate of perspiration from the body decreases.   in other words, the human body feels warmer in humid conditions.   the opposite is true when the relative humidity decreases because the rate of perspiration increases.   the body actually feels cooler in arid conditions.   there is direct relationship between the air temperature and relative humidity and the heat index, meaning as the air temperature and relative humidity increase (decrease), the heat index increases (decreases).

step-by-step explanation: