find [dy/dx] x^2
Using this :
limit as the change of x —-> 0 for
f(x + deltax) - f(x)

run through each step

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  1. f(x+dx) = (x+dx)^2

    = x^2 + 2 x dx + dx^2
    f(x) = x^2
    ------------------------ subtract
    f(x+dx) - f(x) = 2 x dx + dx^2
    divide by dx
    [ f(x+dx) - f(x) ] / dx = 2 x + dx
    let dx ----> 0
    = 2 x
    the end
    d/dx (x^2) = 2 x is now established in your mind forever.

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  2. now if f(x) = x^n
    f(x+dx) = (x+dx)^n = x^n + n x^(n-1) dx + [n(n-1)/2!]x^(n-2) dx*2 ..... + ....dx^bigger
    f(x) = x^n still
    f(x+dx) -f(x) = n x^(n-1)dx+[n(n-1)/2!]x^(n-2) dx*2 .....
    divide by dx
    [f(x+dx) -f(x)] /dx = n x^(n-1) + [n(n-1)/2!]x^(n-2) dx ..... +...dx^ bigger
    let dx ---> 0
    d/dx(x^n) = n x^(n-1)
    so for n = 5173
    d/dx(x^5173) = 5173 x^5172 :)

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