Differential calculus measures how fast one quantity changes when another changes. In biology and the laboratory the interest is usually in a rate: how quickly a culture grows, how fast a tracer leaves a sample, or how steeply a calibration curve rises at a chosen point. The derivative of a function gives that rate at every value of the variable, and the gradient of the tangent to the graph gives the same rate as a slope.
If y depends on x, then a small change in x, written δx, produces a small change in y, written δy. The ratio δy ÷ δx is the average rate of change over that interval, and it is the gradient of the chord joining two points on the graph. When the interval shrinks towards zero, the chord becomes the tangent at the point, and the ratio settles to a fixed value. That value is the derivative of y with respect to x, which is the instantaneous rate of change and the gradient of the curve at that point.
The derivative of y with respect to x can be written in several equivalent ways:
The letter that stands for the independent variable can change: if length L depends on time t, the derivative is dL/dt, with the units of L per unit of t.
A function is a rule that gives exactly one output for each input. The notation f(x) names the function f and shows that x is the input. The output is also written as y, so y = f(x). The input is the independent variable and the output is the dependent variable. Letters can be chosen to suit the subject: H(t) for the height of a seedling at time t, or A(c) for the absorbance at concentration c.
To evaluate a function, replace every x in the rule by the given value, using brackets for negative values and for expressions.
Example: the height of a seedling in a growth chamber, in mm, is modelled by H(t) = 12 + 3t + 0.5t², with t in days.
H(0) = 12 + 0 + 0 = 12 mm
H(4) = 12 + 3(4) + 0.5(16) = 12 + 12 + 8 = 32 mm
H(5) = 12 + 3(5) + 0.5(25) = 12 + 15 + 12.5 = 39.5 mm
The growth during the fifth day is H(5) - H(4) = 39.5 - 32 = 7.5 mm.
Example: for f(x) = 3x² - 2x + 1, evaluate f(-1), f(0) and f(2), and write f(2x).
f(-1) = 3(-1)² - 2(-1) + 1 = 3 + 2 + 1 = 6
f(0) = 0 - 0 + 1 = 1
f(2) = 3(4) - 2(2) + 1 = 12 - 4 + 1 = 9
f(2x) = 3(2x)² - 2(2x) + 1 = 12x² - 4x + 1
A change of input is expressed with f(x + h), where h is a small step. Substitute x + h for x and expand:
f(x + h) = 3(x + h)² - 2(x + h) + 1
f(x + h) = 3x² + 6xh + 3h² - 2x - 2h + 1
This expression is the starting point of differentiation from first principle. The set of inputs for which the rule works is the domain: f(x) = 1 ÷ x is not defined at x = 0, and √x is not defined for negative x when real numbers are used.
Differentiation from first principle finds the derivative directly from the definition, without using any shortcut rule. The gradient of the chord from x to x + h is [f(x + h) - f(x)] ÷ h. The derivative is the limit of this ratio as h approaches zero:
f'(x) = lim (h → 0) [f(x + h) - f(x)] / h
The method always has the same stages: write f(x + h), subtract f(x), divide by h, simplify so that h cancels, and then let h approach zero.
Steps for differentiating from first principle:
Example: the root length of a seedling is L(t) = 2t² + 3t mm, with t in days.
L(t + h) = 2(t + h)² + 3(t + h) = 2t² + 4th + 2h² + 3t + 3h
L(t + h) - L(t) = 4th + 2h² + 3h
[L(t + h) - L(t)] / h = 4t + 2h + 3
As h → 0: L'(t) = 4t + 3
At t = 3 days the root grows at L'(3) = 4(3) + 3 = 15 mm/day. A numerical check at t = 3 shows the chord gradient closing in on 15 as the step shrinks:
h = 1: [L(4) - L(3)] / 1 = (44 - 27) / 1 = 17
h = 0.1: [L(3.1) - L(3)] / 0.1 = (28.52 - 27) / 0.1 = 15.2
h = 0.01: [L(3.01) - L(3)] / 0.01 = (27.1502 - 27) / 0.01 = 15.02
Example: differentiate f(x) = x³ from first principle.
(x + h)³ = x³ + 3x²h + 3xh² + h³
f(x + h) - f(x) = 3x²h + 3xh² + h³
Divide by h: 3x² + 3xh + h²
As h → 0: f'(x) = 3x²
Example: for a constant f(x) = 5, f(x + h) - f(x) = 5 - 5 = 0, so the ratio is 0 for every h, and f'(x) = 0. A constant does not change, so its rate of change is zero.
Differentiating from first principle every time is slow. The same working, done once for each type of function, gives standard results that are used directly. The rules below apply to any x for which the expressions are defined.
Roots and reciprocals are written as powers before differentiating: √x = x^(1/2), 1/x = x⁻¹ and 1/x² = x⁻².
Example: differentiate y = 5x⁴ - 3x² + 7x - 2.
dy/dx = 5(4x³) - 3(2x) + 7(1) - 0
dy/dx = 20x³ - 6x + 7
Example: differentiate y = 4√x + 3/x².
y = 4x^(1/2) + 3x⁻²
dy/dx = 4(1/2)x^(-1/2) + 3(-2)x⁻³
dy/dx = 2x^(-1/2) - 6x⁻³ = 2/√x - 6/x³
Example: differentiate y = 2 sin 3x + cos x.
dy/dx = 2(3 cos 3x) + (-sin x)
dy/dx = 6 cos 3x - sin x
Example: the seedling height in the functional notation example, H(t) = 12 + 3t + 0.5t², has derivative H'(t) = 3 + t mm/day. At t = 4 days, H'(4) = 7 mm/day, close to the 7.5 mm actually gained over the whole fifth day, because the rate is still rising during that day.
Example: the number of cells in a culture is N(t) = 150 e^(0.25t), with t in hours.
dN/dt = 150 × 0.25 × e^(0.25t) = 37.5 e^(0.25t)
At t = 4: N = 150 e¹ = 407.7 cells and dN/dt = 37.5 × e¹ = 101.9 cells per hour
The growth rate, 101.9, equals 0.25 × 407.7, a quarter of the current number of cells, which is the signature of exponential growth.
Example: find the gradient of y = x³ - 4x at x = 2.
dy/dx = 3x² - 4
At x = 2: dy/dx = 3(4) - 4 = 8
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