Solve Partial Derivatives: Find f(x,y)

In summary, the conversation discusses solving a problem involving partial derivatives and finding a function that satisfies certain conditions. The solution involves integrating the given partial derivatives and comparing the resulting expressions to determine the value of the constant term in the function. The final function found is yex + x + sin(y) plus a constant.
  • #1
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[SOLVED] Partial Derivatives

I'm having a bit of trouble on an old test problem. It states:

Determine if there is a function f(x, y) such that fx(x, y) = yex + 1 and fy(x, y) = ex + cos(y). If such a function exists, find it.

I know that such a function exists because fxy(x, y) = ex, and fyx(x, y) = ex, thus fxy = fyx.

I'm having trouble finding the original function. I know that I would probably have to integrate somehow, but I'm not sure how to go about that with partial derivatives.

Any help is appreciated.
 
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  • #2
integrate fx(x, y) with respect to x. This will tell you that f(x,y) is of the form e^x+x+C(y), where C(y) is some function of y. Do the same thing, but with fy(x, y).. you'll get f(x,y)=...+D(x) where CDx) is some function of x. Compare the two expressions of f(x,y) thus obtained, and conclude as to the value of C(y) and D(x), and hence of f(x,y)
 
  • #3
Ok, I found that f(x, y) = yex + x + sin(y), and it checks with the first order partial derivatives. Thank you very much :)
 
  • #4
Plus a constant, of course.
 
  • #5
Right, I forgot that part, haha. Thanks :)
 

FAQ: Solve Partial Derivatives: Find f(x,y)

What is a partial derivative?

A partial derivative is a mathematical concept used to calculate the rate of change of a multivariable function with respect to one of its variables while holding the other variables constant. In simpler terms, it measures how much a function changes when only one of its variables changes, while keeping the other variables fixed.

Why are partial derivatives important?

Partial derivatives are important because they allow us to analyze the behavior of multivariable functions in a certain direction. This is useful in many fields, such as physics, economics, and engineering, where we often encounter functions with multiple variables.

How do you solve for partial derivatives?

To solve for partial derivatives, we use the same rules and techniques as we do for ordinary derivatives. However, we treat all other variables as constants and only differentiate with respect to the variable of interest. This means we can use the power rule, product rule, quotient rule, and chain rule, among others, to find the partial derivative.

What is the difference between partial derivatives and total derivatives?

The main difference between partial derivatives and total derivatives is that partial derivatives only consider the change in one variable while holding the others constant, whereas total derivatives consider the change in all variables simultaneously. In other words, total derivatives take into account the effect of all variables on the function, while partial derivatives only focus on one variable at a time.

How are partial derivatives used in real-world applications?

Partial derivatives have various real-world applications, such as optimization problems in economics, finding maximum and minimum values in engineering, and determining rates of change in physics. They are also used in machine learning and data analysis to identify relationships and patterns between multiple variables.

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