Nib asked
Why is the derivative of ln x 1/x?
The natural log is the mirror image of e to the x across the line y equals x, and mirroring turns a slope into its reciprocal; where e to the x has slope x's mirror value, ln x has slope one over x.
5 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1getting flatter
The natural logarithm ln x grows without bound but slowly. Its slope at x = 1 is 1, at x = 3 it is 1/3 and at x = 10 it is 1/10. In general the slope is 1/x, and it is only defined for x greater than 0.
ln x has slope 1/x: steep near 0, flatter as x grows.
a common mix-upThe slope of ln x is not ln x again or x ln x: it is 1/x, which is why the curve flattens as x grows.
2the proof
Write y = ln x, which means e^y = x. Differentiate both sides with respect to x. The chain rule gives e^y times dy/dx on the left, and 1 on the right. So dy/dx = 1/e^y, and since e^y = x, dy/dx = 1/x.
e^y y' = 1, so y' = 1/e^y = 1/x.
3mirror images
The inverse of a function is its graph reflected over the line y = x. A reflection swaps horizontal and vertical, so a tangent line with slope m becomes a line with slope 1/m.
Reflection over y = x turns slope m into 1/m.
4height 3
On y = e^x, the point (ln 3, 3) has slope e^(ln 3) = 3. Its mirror image is (3, ln 3) on y = ln x, where the slope is the reciprocal, 1/3. Since the x-coordinate there is 3, the slope is 1/x.
The mirror of slope e^y = x is slope 1/x.
What is the slope of ln x at x = 5?
Show the answer
One fifth, since the slope of ln x at x is 1/x.
5the whole idea
The derivative of ln x is 1/x for x > 0. For a logarithm in another base b, log_b x = ln x / ln b, so its derivative is 1/(x ln b).
d/dx ln x = 1/x.
Natural log undoes e to the x, and undoing flips slopes upside down, so its slope at x is one over x.