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Foundations: Maths

Chain rule: derivative of e^(ax + b)

Slope of y = e^(ax + b) at x.

Ref: FMA-64
Rev: v1
Field: Foundations: Maths
Result: dy/dx

\frac{d}{d x} e^{a x + b} = a e^{a x + b}

For y = exp(ax + b) the inside function is u = ax + b, with du/dx = a, and the outside function is exp(u), with derivative eᵘ. The chain rule multiplies the two rates: dy/dx = (outer rate) × (inner rate) = a·e^(ax + b).

Slope of y = e^(ax + b) at x. The outer function e^u is its own derivative, and the inner function ax + b contributes the factor a.

Calculator

Solve for

dy/dxGradientComputed

aCoefficient a

−1010

bConstant b

−1010

xPosition x

−55

Inner function u = ax + b1.7
Outer rate at u5.474
Value of y5.474

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.

Gradient () current value 16.42

xPosition xKey driver

1.51e-61.62e7

bConstant b

4.52e-4219,000

aCoefficient a

−1.51900

At x = 0.4 the curve has slope 16.42. The slope is a times the value of the curve itself.
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	1.513e-6	1.617e7	1.617e7	+0.2	+82.2 %
b Constant b	−10 to 10	4.522e-4	219,400	219,400	+0.4	+49.2 %
a Coefficient a	−10 to 10	−1.513	900.2	901.7	+0.4	+33 %

Sweep of Position x

Position x	Gradient
−5	1.513e-6
−3.75	6.434e-5
−2.5	0.002736
−1.25	0.1163
0	4.946
1.25	210.3
2.5	8,943
3.75	380,300
5	1.617e7

Sweep one variable

Vary

Gradient ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

dy/dx

Gradient

Result ·

What the equation gives you. Everything else feeds it.

Coefficient a

· typical −10 to 10

The coefficient of x inside the bracket. Raises the slope strongly, since a appears as a factor and in the exponent. Raising it by 10 % (to 3.3) moves the result by +24 %, from 16.42 to 20.37.

Constant b

· typical −10 to 10

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 +5.13 %, from 16.42 to 17.26.

Position x

· typical −5 to 5

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 +12.7 %, from 16.42 to 18.52.

Worked example with the values above

Inner function: u = 3 × 0.4 + 0.5 = 1.7, with du/dx = 3.
Outer rate: eᵘ at u = 1.7 gives 5.47395.
dy/dx = 5.47395 × 3 = 16.4218.

Assumptions and limits

The inner and outer functions are differentiable at the chosen point.
Real numbers.

Common mistakes

Forgetting the factor a from the exponent.
Writing e^(ax + b) as e^a · x.
Mixing up e^(ax + b) with a^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

Rate of charging or discharging in an RC circuit.
Growth and decay rates in cooling and population models.

Maths you need

Derivatives: d/dx of sin, eˣ and ln x
Composite functions: a function of a function

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