Partial Derivatives: Evaluating Quotients with Multiple Variables

[patata]

Hey everybody, first time poster although I've recently come across this forum and it's helped me discover the solution of many problems I've been having. I've seen to come to grips with most partial derivative problems I've come across, however, i still can't get correct solutions to problems involving quotients, don't know how exactly to go about it. The two specific questions I've got in my book which i can't seem to get are given below.

Homework Statement

Evaluate the first partial derivative

Homework Equations

f(x, y) = (x + y)/(x - y) (1)

f(x, y) = log(1+x) / log(1+y) (2)

Thanks for any help which could be given =)

I assume you use the standard quotient rule, but i don't know how exactly to go about it with multiple variables...thanks.

 

[Doc Al]

I assume you use the standard quotient rule, but i don't know how exactly to go about it with multiple variables...thanks.

Yes, you'd use the quotient (or product) rule and the chain rule. When computing ∂F/∂x, treat y as a constant; When computing ∂F/∂y, treat x as a constant.

 

[Pengwuino]

Yes, you simply use the quotient rule, keeping the variable you're not taking the partial derivative of as a constant.

$$\begin{array}{l} f(x,y) = \frac{{g(x,y)}}{{h(x,y)}} \\ \frac{{\partial f(x,y)}}{x} = \frac{{\frac{{\partial g(x,y)}}{{\partial x}}h(x,y) - g(x,y)\frac{{\partial h(x,y)}}{{dx}}}}{{h(x,y)^2 }} \\ \end{array}$$

is an example of the partial with respect to x where you already know how to properly take partial derivatives of functions.

 

[patata]

Thanks for the help guys but I am still getting something wrong, which I am guessing is a small but stupid mistake...

for df/dx i get 1/(1+x) . -1(log(1+y))^-2 (unless that's right?)

for df/dy I am confused...whats the derivative of a log of a constant :S

Thanks for any help =)

 

[chislam]

Well the derivative of a constant is 0. And the log of a constant is just another number.