How Does the Chain Rule Relate ∂h/∂z to ∂g/∂x?

In summary, the conversation discusses finding the partial derivative of h(y,z) for z given an implicit function f(x,y,z)=0. The speaker knows the partial derivative of g(x,y) for x and wants to determine the partial derivative of h(y,z) for z. The conversation suggests using the chain rule to relate the two partial derivatives.
  • #1
cruxcriticoru
1
0
If given an implicit function f(x,y,z)=0. Then, we can get z=g(x,y) and x= h(y,z).
I know the answer for the partial derivative of g(x,y)' for x, how can I know the partial derivative of h(y,z) for z?

I know f( x, y, z)=0. And I know ∂g / ∂x is positive.
How can I define whether ∂h / ∂z is negative or positive?
How can I express ∂h / ∂z in terms of ∂g / ∂x?
 
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  • #2
Hi cruxcriticoru, welcome to PF.

Probably your answer involves the chain rule. You know that ∂g / ∂x = ∂z / ∂x is positive. Can you relate ∂h / ∂z with ∂z / ∂x using the chain rule?
 

Related to How Does the Chain Rule Relate ∂h/∂z to ∂g/∂x?

1. What is a partial derivative?

A partial derivative is a mathematical concept used in multivariate calculus to measure the rate of change of a function with respect to one of its variables, while holding all other variables constant. It allows us to analyze how a function changes in relation to specific variables, rather than all variables at once.

2. How do you calculate a partial derivative?

To calculate a partial derivative, you first identify the variable that is being held constant and treat it as a constant in the function. Then, you differentiate the function with respect to the variable of interest, while treating all other variables as constants. This results in a new function that represents the rate of change of the original function with respect to the chosen variable.

3. What is the difference between a partial derivative and a total derivative?

A partial derivative measures the rate of change of a function with respect to one of its variables, while holding all other variables constant. On the other hand, a total derivative measures the overall rate of change of a function with respect to all of its variables. In other words, a partial derivative only considers changes in one variable, while a total derivative considers changes in all variables simultaneously.

4. What are some real-world applications of partial derivatives?

Partial derivatives are commonly used in physics, engineering, economics, and other sciences to model and analyze complex systems. They can be used to optimize processes, such as finding the maximum or minimum value of a function, or to understand how a system will respond to changes in different variables.

5. Can you give an example of a partial derivative problem?

Sure, here is a simple example: Let's say we have a function f(x,y) that represents the surface area of a cube with side lengths x and y. To find the rate of change of the surface area with respect to x, we would calculate the partial derivative ∂f/∂x = 2y. This means that for every unit increase in x, the surface area will increase by 2y square units, while holding y constant.

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