Library / Foundations: Maths / FMA-26
Foundations: Maths
Chain rule (power of a linear function)
Gradient of y = (ax + b)ⁿ at a point.
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- FMA-26
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- v1
- Field
- Foundations: Maths
- Result
- dy/dx
u = ax + b is the inner function; du/dx = a. The general rule is d(uⁿ)/dx = n uⁿ⁻¹ du/dx.
To differentiate a function of a function, differentiate the outer power, then multiply by the derivative of what is inside the bracket.
Calculator
- Value y = (ax + b)ⁿ1
- Inner function ax + b1
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.
- Forgetting the factor a is the commonest mistake: the inner derivative must be included.
- The tangent at x has this gradient and passes through the point (x, y).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Position x | 0 to 3 | 12 | 12,000 | 11,990 | +10% | 0 % |
| b Constant b | 0.5 to 3 | 1.5 | 324 | 322.5 | +10% | +33.1 % |
| a Coefficient a | 1 to 6 | 4 | 24 | 20 | +10% | +10 % |
| n Power n | 2 to 6 | 6 | 18 | 12 | +10% | +10 % |
Sweep of Position x
| Position x | Gradient |
|---|---|
| 0 | 12 |
| 0.375 | 115.1 |
| 0.75 | 411.9 |
| 1.125 | 1,005 |
| 1.5 | 1,997 |
| 1.875 | 3,489 |
| 2.25 | 5,586 |
| 2.625 | 8,389 |
| 3 | 12,000 |
Sweep one variable
Variables and how each one contributes
Gradient
What the equation gives you. Everything else feeds it.
Coefficient a
Multiplier of x inside the bracket. Appears as a factor and inside the power.
Constant b
Constant inside the bracket. Raises the base of the power.
Position x
The x value where the gradient is wanted. Raises the base and so the gradient.
Power n
The power. Raises the gradient steeply if the base exceeds 1.
Worked example with the values above
- Inner function: ax + b = 3 × 0 + 1 = 1.
- Outer derivative: n(ax+b)n−1 = 4 × 1.
- Multiply by a: dy/dx = 12.
Assumptions and limits
- A real power n and positive inner value.
- Differentiating with respect to x.
Common mistakes
- Leaving out the derivative of the inner function.
- Reducing the power but forgetting to bring n down.
Statistics and uncertainty
Powers amplify errors: the gradient depends on the inner value to the power n − 1, so a 1% error in ax + b gives about (n − 1)% error in the gradient.
Where it is used
- Gradient of a response curve at an operating point.
- Rates of change in composite models.
Maths you need
- Power rule: d(xⁿ)/dx = nxⁿ⁻¹
- Functions: inner and outer