chmate said:Homework Statement
If [itex]z=\frac{1}{x}[f(x-y)+g(x+y)][/itex], prove that [itex]\frac{\partial }{\partial x}(x^2\frac{\partial z}{\partial x})=x^2\frac{\partial^2 z}{\partial y^2}[/itex]
Homework Equations
The Attempt at a Solution
I don't know how I'm supposed to find the partial derivative of z with respect to any of variables if the function f and g are not expressed.
Help me!
Thank you
LCKurtz said:Just leave the f and g in there, maybe they will cancel out. For example, if you want the partial of f(x-y) with respect to y it would be:$$
f_y(x-y) = -1\cdot f'(x-y)$$and so on.
chmate said:Can you explain me why [itex]f_y(x-y) = -1\cdot f'(x-y)[/itex]? I don't get it.
What is then [itex]f_x(x+y) = ?[/itex]
A partial derivative is a mathematical concept used to measure 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 often used in multivariable calculus and physics.
Finding the partial derivatives of a function allows us to analyze how the function changes in different directions. This information is essential in many fields, such as economics, engineering, and physics, where functions often depend on multiple variables and their rates of change.
To find the partial derivatives of a function, we use the same rules and techniques as finding regular derivatives. However, we treat all other variables as constants and differentiate the function with respect to the variable of interest. This process is repeated for each variable in the function.
The main difference between partial derivatives and regular derivatives is that partial derivatives measure the rate of change of a function along a specific direction, while regular derivatives measure the overall rate of change of a function. Additionally, partial derivatives involve treating other variables as constants, whereas regular derivatives do not.
Partial derivatives have numerous real-life applications, including optimizing production processes in manufacturing, determining the direction of steepest ascent in economics, and predicting the trajectory of objects in physics. They are also used in machine learning and data analysis to understand the relationships between multiple variables.