User:GFauxPas

Alternate proof of power rule for real indices
Very rough draft, I'm just getting the skeleton of the proof down.

let y = x^n, n is a constant

If we take the natural log of both sides,

ln y = ln x^n

Using the properties of logarithms

ln y = n ln x

Note that the natural log has the same domain as y. Also, both log and x^n are differentiable everywhere. We are then justified in taking the derivative of both sides with respect to x

1/y * dy/dx = n(1/x)

dy/dx = ny/x

dy/dx = n(x^n)/x

dy/dx = nx^(n-1)

QED

Copy paste
For my own benefit


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