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.
Key Formula
d/dx[xⁿ] = n · xⁿ⁻¹
LaTeX: \frac{d}{dx}[x^n] = nx^{n-1}
| Symbol | Meaning | Unit |
|---|---|---|
| x | independent variable | dimensionless |
| n | exponent (any real number) | dimensionless |
| d/dx | differentiation operator with respect to x | dimensionless |
Worked Example
Problem
Differentiate f(x) = 4x³ − 7x² + 2x − 9.
Solution
Step 1: Differentiate each term separately using the power rule d/dx[xⁿ] = nxⁿ⁻¹. Step 2: d/dx[4x³] = 4 · 3x² = 12x². Step 3: d/dx[−7x²] = −7 · 2x = −14x. Step 4: d/dx[2x] = 2 · 1 = 2. Step 5: d/dx[−9] = 0 (constant rule). Step 6: Combine all terms.
Answer
f'(x) = 12x² − 14x + 2
Key Differentiation Rules and Their Formulae
| Rule Name | Formula | When to Use | Example |
|---|---|---|---|
| Power Rule | d/dx[xⁿ] = nxⁿ⁻¹ | Single power of x | d/dx[x⁴] = 4x³ |
| Sum/Difference Rule | d/dx[f ± g] = f' ± g' | Sum or difference of functions | d/dx[x² + x] = 2x + 1 |
| Constant Multiple Rule | d/dx[cf] = c·f' | Function multiplied by constant | d/dx[5x³] = 15x² |
| Product Rule | d/dx[fg] = f'g + fg' | Product of two functions | d/dx[x·sin x] |
| Chain Rule | d/dx[f(g(x))] = f'(g(x))·g'(x) | Composition of functions | d/dx[sin(x²)] |
| Quotient Rule | d/dx[f/g] = (f'g − fg')/g² | Ratio of two functions | d/dx[x/sin x] |
Interactive Tools
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Related Terms
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.
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.
Product Rule (differentiation)
The product rule states that the derivative of the product of two differentiable functions equals the first function times the derivative of the second, plus the second function times the derivative of the first. It is a fundamental rule of differential calculus that prevents the incorrect assumption that the derivative of a product is simply the product of the derivatives. The product rule is used whenever functions are multiplied together and their rate of change is needed, for example in physics when computing power as the product of force and velocity.
From the Latin "differentia" meaning difference. The term "differentiation" as a calculus process was established in the 17th and 18th centuries through the work of Leibniz and Newton. Leibniz introduced the differential notation d(y)/d(x) around 1675.