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ck00
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let f(x,y)=0
Why df=(∂f/∂x)dx + (∂f/∂y)dy?
Why df=(∂f/∂x)dx + (∂f/∂y)dy?
HallsofIvy said:If x and y are themselves functions of a parameter, say, t, then we can think of f(x, y)= f(x(t), y(t)) as a function of the single variable t and, by the chain rule:
[tex]\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}[/tex]
And then, the usual definition of the "differential" as df= (df/dt)dt gives
[tex]df= \frac{\partial f}{\partial x}\frac{dx}{dt}dt+ \frac{\partial f}{\partial y}\frac{dy}{dt}dt= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y} dy[/tex]
ck00 said:But how can I prove this [tex]\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}[/tex]?
I know the basic operation in partial differentiation but i just not quite understand the theory behind it. Are there some proofs?
let f(x,y)=0
Why df=(∂f/∂x)dx + (∂f/∂y)dy?
The notation df=(∂f/∂x)dx + (∂f/∂y)dy represents the total differential of a multivariable function f. It indicates the change in the function f due to small changes in the variables x and y.
The partial derivative (∂f/∂x)dx represents the change in the function f with respect to the variable x, while keeping all other variables constant. It is used in the equation to account for the change in the function due to changes in x.
(∂f/∂y)dy and (∂f/∂x)dx are both partial derivatives, but with respect to different variables. (∂f/∂y)dy represents the change in f with respect to y, while keeping all other variables constant. It is different from (∂f/∂x)dx, which represents the change in f with respect to x, while keeping all other variables constant.
Sure, let's say we have a function f(x,y) = x^2 + y^2. To find the change in f when x increases by 1 and y increases by 2, we can use the equation df=(∂f/∂x)dx + (∂f/∂y)dy. Plugging in the values, we get df = (2x)(dx) + (2y)(dy). Since x increases by 1 and y increases by 2, dx = 1 and dy = 2. Therefore, df = (2)(1) + (2)(2) = 6. So, the change in f is 6 when x increases by 1 and y increases by 2.
This equation is commonly used in fields such as physics, engineering, economics, and statistics where multivariable functions are used to model real-world phenomena. It is particularly useful in calculating rates of change and optimization problems.