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Math Help - Partial Derivatives: What's the basis for the general algorithm?

  1. #1
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    Partial Derivatives: What's the basis for the general algorithm?

    I'm trying to understand why you keep certain variables constant when taking partial derivatives. For example, for a function f(x,y), to take the partial derivative of f with respect to y, you "consider" x a constant and take the normal derivative with respect to 'y'.

    In single variable calculus, the underlying idea of derivatives involve limits. The algorithm, to which I generally refer for example, would be "nx^(n-1)" for the derivative of x^n.

    I'm hoping that if I know the reason behind the algorithm, I can understand what it geometrically means when keeping other variables constant.
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    Re: Partial Derivatives: What's the basis for the general algorithm?

    Hi Elusive. The idea of taking partial derivatives is simple and intuitive. When you have a function of multiple independent variables (say 2), there are many occasions when you would like to find what is the slope of the function in a particular direction. The partial derivative with respect to x signifies the slope of the function in the x-direction and the one with y signifies the slope in the y direction. Unlike 1D, in 2D, f(x,y) which is a surface has infinite number of tangent lines at any point and each one of them signifies the rate of change of the function in that direction.

    Now lets compare this to the total derivative. This is usually used when the 2 variables x and y are not independent. For example, say you have some z which is a function of the position of a particle (in the horizontal direction i.e x) and its elevation (y) and suppose your elevation y changes with x in a certain way (i.e you know the variation of altitude along x). In this case x and y are not 2 independent variables and finding the partial derivative w.r.t x does not make sense since you know y changes when x changes. In such cases, you take the total derivative i.e \frac{df}{dx} =  \frac{\partial f}{\partial x} +  \frac{\partial f}{\partial y} \frac{dy}{dx}. I hope that makes things clearer for you.
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