# Simple Partial Derivatives Question

• apw235
In summary: The chain rule is a powerful tool for finding derivatives and can make seemingly complicated problems very simple. In this case, yes, you would use 0.15 for dT/dt and 0.1 for dR/dt. In summary, the given function for wheat production, W(T,R), is affected by changes in temperature and rainfall. The partial derivatives ∂W/∂T and ∂W/∂R have different signs, indicating that as temperature increases, wheat production decreases, and as rainfall decreases, wheat production increases. Using the chain rule, the current rate of change of wheat production, dW/dt, can be calculated by multiplying the partial derivatives by the corresponding rates of change for temperature and rainfall.
apw235

## Homework Statement

Wheat production in a given year, W, depends on the average temperature T and the annual rainfall R. Scientists estimate that the average temperature is rising at a rate of 0.15 degrees celsius per year and rainfall is decreasing at a rate of 0.1 cm per year. The also estimate that, at current production levels, ∂W/∂T = -2 and ∂W/∂R = 8.

a) what is the significance of the signs of the partials?
b) estimate the current rate of change of wheat production dW/dt.

## The Attempt at a Solution

So the function is W(T,R). taking the partial with respect to temperature and getting a negative value means that as the temperature is rising, Wheat production is decreasing, and the positive value for rainfall decreasing means that wheat production is rising as rainfall decreases? this doesn't make sense to me, why would the wheat production increase as rainfall decreased...am i thinking about it wrong?

i'm not sure what they want us to do for part b though.., haven't tried that part yet.

thanks

Chain rule for partial derivatives. dW/dt=(dW/dR)*(dR/dt)+(dW/dT)*(dT/dt). Look it up. You'll need to substitute partial symbols for some of those d's.

i know the chain rule but do i use .15 as dT/dt and .1 as dR/dt? then it becomes extremely trivial, i thought it would be more involved...oh well

It's not.

## 1. What is a simple partial derivative?

A simple partial derivative is a mathematical tool used to measure how a function changes as one of its variables changes, while holding all other variables constant. It represents the rate of change of a function with respect to one variable.

## 2. How do I calculate a simple partial derivative?

To calculate a simple partial derivative, you take the derivative of the function with respect to the variable of interest, treating all other variables as constants. This is done using the standard rules of differentiation, such as the power rule or product rule.

## 3. What is the purpose of using simple partial derivatives?

Simple partial derivatives are used to understand the behavior of a function with respect to a single variable. They are often used in optimization problems, where the goal is to find the maximum or minimum value of a function.

## 4. How is a simple partial derivative different from a total derivative?

A simple partial derivative only considers the change in a function with respect to one variable, while a total derivative takes into account the changes in all variables. This means that a simple partial derivative is a special case of a total derivative, where all other variables are held constant.

## 5. Can simple partial derivatives be negative?

Yes, simple partial derivatives can be negative. This indicates that the function is decreasing in the direction of the variable of interest. A positive simple partial derivative indicates that the function is increasing in that direction.

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