Derivatives - Measuring Rates of Change

A derivative measures how fast a function is changing at any given point. Think of it as the instantaneous rate of change – not the average change over a long interval, but the exact rate at one specific instant. If a function describes a position, its derivative describes velocity. If the function describes velocity, its derivative describes acceleration.

The derivative was discovered independently by Isaac Newton in England, who called it a “fluxion” and developed it around 1666 to describe motion, and Gottfried Wilhelm Leibniz in Germany, who published his own version in 1684 using the dy/dx notation still used today. The two men never resolved who deserved credit first, and the resulting calculus priority dispute split European mathematics for decades – British mathematicians stubbornly stuck with Newton's clunkier fluxion notation well into the 19th century largely out of national pride, which is one reason continental Europe pulled ahead in mathematical physics during that period.

The Gradient of a Curve

On a straight line, the gradient (slope) is constant and easy to compute. On a curve, the gradient is different at every point. The derivative at point x gives the gradient of the tangent line to the curve at that exact point. The tangent line just touches the curve at one point and has the same slope as the curve there.

The Formal Definition

The derivative of f at x is defined as the limit:
f′(x) = lim_{h → 0} [f(x + h) − f(x)] / h
This is called the first principles definition. It calculates the slope of a secant line as the two points move infinitely close together.

Notation

The derivative of y = f(x) can be written as:
f′(x) or dy/dx or y′ or d/dx [f(x)]
All four mean exactly the same thing.

Differentiation Rules

Rule	Formula	Example
Power Rule	d/dx [xⁿ] = nxⁿ⁻¹	d/dx [x⁵] = 5x⁴
Constant Rule	d/dx [c] = 0	d/dx [7] = 0
Constant Multiple	d/dx [cf(x)] = cf′(x)	d/dx [3x²] = 6x
Sum / Difference	d/dx [f ± g] = f′ ± g′	d/dx [x³ + x] = 3x² + 1
Product Rule	d/dx [fg] = f′g + fg′	d/dx [x² sin x] = 2x sin x + x² cos x
Quotient Rule	d/dx [f/g] = (f′g − fg′) / g²	d/dx [sin x / x] = (x cos x − sin x) / x²
Chain Rule	d/dx [f(g(x))] = f′(g(x)) · g′(x)	d/dx [(x²+1)³] = 3(x²+1)² · 2x

Derivatives of Standard Functions

Function	Derivative
sin x	cos x
cos x	−sin x
tan x	sec² x
e^x	e^x
ln x	1/x
xⁿ	nxⁿ⁻¹

Worked Examples

Differentiate f(x) = 4x³ − 7x² + 2x − 5.

Apply the power rule term by term:
f′(x) = 12x² − 14x + 2.

Find the gradient of y = x³ − 3x at x = 2.

dy/dx = 3x² − 3. At x = 2: 3(4) − 3 = 12 − 3 = 9.

Differentiate y = (2x + 1)⁴ using the chain rule.

Outer function: u⁴, derivative = 4u³. Inner function: 2x + 1, derivative = 2.
dy/dx = 4(2x + 1)³ × 2 = 8(2x + 1)³.

Find where y = x³ − 6x² + 9x has a stationary point.

dy/dx = 3x² − 12x + 9. Set equal to zero:
3(x² − 4x + 3) = 0 → 3(x − 1)(x − 3) = 0.
Stationary points at x = 1 and x = 3.

Second Derivative

The second derivative f″(x) or d²y/dx² is the derivative of the derivative. It measures how the rate of change is itself changing – in other words, the concavity of the curve.
If f″(x) > 0 at a stationary point: local minimum.
If f″(x) < 0 at a stationary point: local maximum.

Key Takeaways

The derivative gives the instantaneous rate of change and the slope of the tangent line.
Power rule: d/dx [xⁿ] = nxⁿ⁻¹.
Product, quotient, and chain rules handle more complex expressions.
Stationary points occur where f′(x) = 0; the second derivative classifies them.

Practice: Gradients and Stationary Points

Finding a Gradient