Partial Derivatives of f(x,y) = (4x+2y)/(4x-2y) at (2,1) - Step-by-Step Guide

Thread starter mstange89
Start date Mar 6, 2010
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Derivative Partial Partial derivative

In summary, a partial derivative is a mathematical concept used to measure the rate of change of a function with respect to one variable while holding all other variables constant. To calculate a partial derivative, all other variables are treated as constants and the function is differentiated as normal. Partial derivatives have various applications in mathematics and science, including optimization and problem-solving. An example of calculating a partial derivative is shown for the function f(x,y) = (4x+2y)/(4x-2y) at the point (2,1). The interpretation of a partial derivative is that it represents the rate of change of the function in a specific direction at a specific point. A positive value indicates an increasing function, a negative value indicates a decreasing function,

Mar 6, 2010

mstange89

Partial Derivative help!

Find the first partial derivatives of f(x,y)=(4x+2y)/(4x−2y) at the point (x,y) = (2, 1).

My professor never taught us how to do this so I have no idea where to start. Any help would be appreciated.

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Mar 6, 2010

Hurkyl

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Science Advisor

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Er... you weren't taught about derivatives?

Mar 6, 2010

mstange89

Derivatives yes, partial derivatives with a quotient, no

FAQ: Partial Derivatives of f(x,y) = (4x+2y)/(4x-2y) at (2,1) - Step-by-Step Guide

What is a partial derivative?

A partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables while holding all other variables constant. It is denoted by ∂ (pronounced "partial") and is used in multivariable calculus to study the behavior of functions in multiple dimensions.

How do you calculate a partial derivative?

To calculate a partial derivative, you first treat all other variables except the one you are differentiating with respect to as constants. Then, you differentiate the function as you normally would with a single variable. For example, to find the partial derivative of f(x,y) with respect to x, you would treat y as a constant and differentiate f(x,y) with respect to x.

What is the purpose of calculating partial derivatives?

Partial derivatives are useful in many areas of mathematics and science, including economics, physics, and engineering. They allow us to understand how a function changes in different directions and can help us optimize functions and solve problems involving multiple variables.

Can you provide an example of calculating a partial derivative?

Sure! Let's take the function f(x,y) = (4x+2y)/(4x-2y) and find the partial derivative with respect to x at the point (2,1).

First, we treat y as a constant and differentiate the function with respect to x:

∂f/∂x = (4(4x-2y) - (4x+2y)(4))/(4x-2y)^2

Next, we plug in the point (2,1) for x and y:

∂f/∂x = (4(4(2)-2(1)) - (4(2)+2(1))(4))/(4(2)-2(1))^2

∂f/∂x = 0

Therefore, the partial derivative of f(x,y) with respect to x at the point (2,1) is 0.

How can we interpret the value of a partial derivative?

The value of a partial derivative at a specific point tells us the rate of change of the function with respect to the variable we are differentiating with respect to at that point. In other words, it tells us how much the function is changing at that point in a specific direction. A positive value indicates an increasing function, while a negative value indicates a decreasing function. A value of 0 indicates that the function is not changing in that direction at that point.