Derivative (calculus)
The derivative of a function at a point measures the instantaneous rate of change of the function's output with respect to its input at that point, and geometrically represents the slope of the tangent line to the function's graph. Derivatives are defined as the limit of the difference quotient as the interval shrinks to zero. They are central to physics, engineering, economics, and all sciences wherever rates of change or optimisation are relevant.
Key Formula
f'(x) = lim (h → 0) [f(x+h) − f(x)] / h
LaTeX: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
| Symbol | Meaning | Unit |
|---|---|---|
| f'(x) | derivative of f at x | units of f / units of x |
| h | small increment in x approaching zero | same as x |
| f(x+h) | value of function at x + h | same as f(x) |
| f(x) | value of function at x | same as f(x) |
Worked Example
Problem
Using the limit definition, find the derivative of f(x) = x² at x = 3.
Solution
Step 1: Write the difference quotient: [f(3+h) − f(3)] / h = [(3+h)² − 9] / h. Step 2: Expand (3+h)²: 9 + 6h + h². Step 3: Subtract 9: (9 + 6h + h² − 9)/h = (6h + h²)/h. Step 4: Factor: h(6 + h)/h = 6 + h. Step 5: Take the limit as h → 0: 6 + 0 = 6.
Answer
f'(3) = 6 (the slope of the tangent to x² at x = 3 is 6)
Standard Derivative Rules for Common Functions
| Function f(x) | Derivative f'(x) | Rule Used | Example |
|---|---|---|---|
| xⁿ | nxⁿ⁻¹ | Power rule | d/dx[x⁵] = 5x⁴ |
| sin x | cos x | Trig rule | d/dx[sin x] = cos x |
| cos x | −sin x | Trig rule | d/dx[cos x] = −sin x |
| eˣ | eˣ | Exponential rule | d/dx[eˣ] = eˣ |
| ln x | 1/x | Log rule | d/dx[ln x] = 1/x |
| c (constant) | 0 | Constant rule | d/dx[7] = 0 |
Interactive Tools
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Related Terms
Limit (calculus)
A limit describes the value that a function approaches as its input approaches a given point, even if the function is not defined at that point. Limits are the foundational concept of calculus, underpinning the rigorous definitions of derivatives and integrals. They are essential for analysing the behaviour of functions near discontinuities, at infinity, and for understanding rates of change.
Differentiation
Differentiation is the process of computing the derivative of a function, yielding a new function that expresses the rate of change of the original at every point in its domain. It involves applying systematic rules — such as the power rule, product rule, chain rule, and quotient rule — to transform a given function into its derivative. Differentiation is used extensively in physics for velocity and acceleration, in economics for marginal analysis, and in engineering for optimisation and control systems.
Chain Rule
The chain rule is a differentiation rule used to compute the derivative of a composite function, stating that the derivative of f(g(x)) equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. It is one of the most widely applied rules in calculus, essential whenever a function is "nested" inside another. The chain rule is critical in physics for related rates problems, in machine learning for backpropagation, and in multivariable calculus for total derivatives.
From the Latin "derivare" meaning to draw off or lead away. The concept was independently developed by Isaac Newton (who called it a "fluxion") and Gottfried Wilhelm Leibniz in the 1660s–1680s. Leibniz's notation dy/dx is the one most commonly used today.