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why is the derivative of ln x 1/x
The graph of natural log keeps climbing but gets flatter. At x equals one its slope is one. At three, a third. At ten, a tenth. Slope is one over x. Why?
Here is the proof. y equals ln x means e to the y equals x. Differentiate both sides: e to the y times y prime equals one. So y prime is one over e to the y, which is one over x.
Now see it. The graphs of e to the x and ln x are mirror images across the line y equals x. A mirror swaps rise and run, so every slope flips into its reciprocal.
Take the point on e to the x where the height is three. There the slope equals the height, three. Its mirror sits at x equals three on ln x, with slope one third.
So the derivative of ln x is one over x, for x above zero. It works because ln undoes e to the x. For log base ten, divide by ln ten as well.
Natural log undoes e to the x, and undoing flips slopes upside down, so its slope at x is one over x.