Understanding Derivatives in Calculus — Rules, Formulas & Applications
Blog › Math · 12 min read · Published 2026-03-12
Master derivatives from first principles to advanced rules. Learn power rule, chain rule, product rule with worked examples and real-world applications.
What Is a Derivative?
A derivative measures the instantaneous rate of change of a function. Geometrically, it represents the slope of the tangent line to a curve at any given point. If f(x) describes position over time, f'(x) gives velocity.
The derivative is defined as: f'(x) = lim(h to 0) [f(x+h) - f(x)] / h
Essential Derivative Rules
Power Rule
d/dx [x^n] = n times x^(n-1). Examples: d/dx[x cubed] = 3x squared, d/dx[sqrt(x)] = 1/(2 sqrt(x)).
Product Rule
d/dx [f(x) times g(x)] = f'(x) times g(x) + f(x) times g'(x). "First times derivative of second, plus second times derivative of first."
Quotient Rule
d/dx [f(x)/g(x)] = [f'(x) times g(x) - f(x) times g'(x)] / [g(x)] squared.
Chain Rule
d/dx [f(g(x))] = f'(g(x)) times g'(x). Used for composite functions like sin(x squared).
Common Derivatives Table
- Constants: d/dx[c] = 0
- Exponential: d/dx[e^x] = e^x
- Logarithmic: d/dx[ln(x)] = 1/x
- Trigonometric: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x)
Worked Examples
Example 1: Polynomial
Find d/dx[3x^4 - 2x^2 + 7x - 5]. Apply power rule: 12x^3 - 4x + 7
Example 2: Chain Rule
Find d/dx[(2x + 3)^5]. Result: 10(2x + 3)^4
Applications of Derivatives
- Physics: Velocity, acceleration, force
- Economics: Marginal cost, marginal revenue, elasticity
- Engineering: Rate of heat flow, circuit analysis
- Optimization: Finding maximum profit, minimum cost
- Machine learning: Gradient descent uses derivatives to minimize loss functions
Higher-Order Derivatives
The second derivative f''(x) measures how the rate of change itself changes. The second derivative test determines whether critical points are maxima (f'' less than 0) or minima (f'' greater than 0).
FAQ
When is a function not differentiable?
At discontinuities, sharp corners (like |x| at x=0), vertical tangents, or infinite oscillations. Differentiability implies continuity, but not vice versa.
How are derivatives used in machine learning?
Gradient descent computes partial derivatives of a loss function to adjust model parameters. Backpropagation is the chain rule applied across neural network layers.