Library / Foundations: Maths / FMA-65
Foundations: Maths
Chain rule: derivative of ln(ax + b)
Slope of y = ln(ax + b) at x.
- Ref
- FMA-65
- Rev
- v1
- Field
- Foundations: Maths
- Result
- dy/dx
For y = ln(ax + b) the inside function is u = ax + b, with du/dx = a, and the outside function is ln(u), with derivative 1/u. The chain rule multiplies the two rates: dy/dx = (outer rate) × (inner rate) = a/(ax + b).
Slope of y = ln(ax + b) at x. The outer natural log gives 1 over its argument and the inner function contributes the factor a. It needs ax + b to be positive.
Calculator
- Inner function u = ax + b1.7
- Outer rate at u0.5882
- Value of y0.5306
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At x = 0.4 the curve has slope 1.765. The slope is a divided by the inside value, so it falls as the curve rises.
- The inner factor a scales every slope: doubling a doubles the slope at a given inner value, and it also moves the point on the curve where the slope is taken.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Position x | −5 to 5 | 0.1935 | 6 | 5.806 | +0.2 | −26.1 % |
| b Constant b | −10 to 10 | 0.2679 | 5.625 | 5.357 | +0.4 | −19 % |
| a Coefficient a | −10 to 10 | −2.857 | 2.222 | 5.079 | +0.4 | +3.58 % |
Sweep of Position x
| Position x | Gradient |
|---|---|
| −5 | — |
| −3.75 | — |
| −2.5 | — |
| −1.25 | — |
| 0 | 6 |
| 1.25 | 0.7059 |
| 2.5 | 0.375 |
| 3.75 | 0.2553 |
| 5 | 0.1935 |
Sweep one variable
Variables and how each one contributes
Gradient
What the equation gives you. Everything else feeds it.
Coefficient a
The coefficient of x inside the bracket. Changes the slope through the factor a and through the angle or argument ax + b. Raising it by 10 % (to 3.3) moves the result by +2.75 %, from 1.765 to 1.813.
Constant b
The constant added inside the bracket. Shifts the point on the curve at which the slope is taken, so the effect depends on the shape of the curve there. Raising it by 10 % (to 0.55) moves the result by −2.86 %, from 1.765 to 1.714.
Position x
The position at which the slope is found. Moves along the curve; the effect on the slope depends on the shape of the curve there. Raising it by 10 % (to 0.44) moves the result by −6.59 %, from 1.765 to 1.648.
Worked example with the values above
- Inner function: u = 3 × 0.4 + 0.5 = 1.7, with du/dx = 3.
- Outer rate: 1/u at u = 1.7 gives 0.588235.
- dy/dx = 0.588235 × 3 = 1.76471.
Assumptions and limits
- The inner and outer functions are differentiable at the chosen point.
- ax + b is positive.
Common mistakes
- Forgetting the factor a in the numerator.
- Evaluating the formula where ax + b is zero or negative (undefined).
- Differentiating ln(ax + b) as 1/x.
Statistics and uncertainty
A relative error in a enters twice here, through the factor a and through ax + b, so slopes calculated from fitted coefficients can carry larger relative errors than the coefficients themselves.
Where it is used
- Rates of change on a logarithmic scale.
- Sensitivity of a logarithmic reading, such as pH or dB, to its input.
Maths you need
- Derivatives: d/dx of sin, eˣ and ln x
- Composite functions: a function of a function