Logarithmic Differentiation Workbench | MIT David Jerison Method

Standard Space: y(x) Slope y'(2) = 6.77

Logarithmic Space: ln(y) Slope d/dx[ln y] = 1.69

Evaluation Point (x)

2.0000

Log Space Side d/dx[ln y]

1.6931

Exact Tangent Slope y'(x)

6.7726

David Jerison Method Intuition

Directly differentiating power towers like x^x is impossible with standard power or exponential rules alone because both base and exponent vary. By applying ln(y), power relationships turn into products, and rational quotients turn into subtractions. The derivative in logarithmic space is simply y'/y, meaning the true slope is obtained by scaling the log-space slope back up by y.

Logarithmic differentiation reveals relative slope

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In the default preset y equals x to the x. Taking the natural logarithm turns the exponent into a multiplier: log y equals x log x. Differentiating gives y prime divided by y equals one plus log x. Multiplying back by y gives y prime equals x to the x times one plus log x. At x equals two, y is four and the slope is about six point seven seven three. The tangent drawn here uses that slope. The slope of log y is y prime divided by y. For x to the x at two, that relative slope is one plus log two, about one point six nine three one four seven. Multiplying it by y equals four recovers the ordinary slope six point seven seven two five eight eight. These slopes are different quantities; the two plots show ordinary y space and logarithmic space side by side. Real logarithms require positive y. Choose the x to the one over x preset. Log y is log x divided by x, so the relative derivative is one minus log x divided by x squared. It is positive before x equals e, zero at e, and negative afterward. The peak is e to the one over e, about one point four four four six six eight. The moving point follows this actual function through x equals two, then e, then three point five.