Derivative Calculator
Differentiate Any Function of x
Differentiate polynomials, trigonometric, exponential and logarithmic functions instantly — with full step-by-step working showing exactly which rule was applied at every stage. Built for TN Board, CBSE, JEE and NEET students.
Enter a function of x
Differentiation Rules Reference
How to Differentiate a Function
The derivative of a function f(x), written f'(x) or dy/dx, measures how fast f(x) changes as x changes — its instantaneous rate of change or slope at any point. Differentiation is built from a small set of core rules that combine to handle any function you'll meet in school or entrance exams.
f'(x) = limh→0 [f(x+h) − f(x)] / h
That limit definition is where every differentiation rule comes from, but in practice you apply the Power Rule, Product Rule, Quotient Rule and Chain Rule directly instead of returning to first principles each time. This calculator shows exactly which of those rules applies at each step of your function, in the order a student would apply them by hand.
A useful check after differentiating: for a polynomial, the derivative's degree should always be exactly one less than the original — differentiating x³ should give something of degree 2, not degree 3 or 1. This calculator's step-by-step trace lets you confirm you applied each rule correctly, not just whether the final answer matches.
← Back to Maths homework helpDerivative Calculator — Questions Answered
Polynomials (x², x³, etc.), sin, cos, tan, exp (eˣ), ln (natural log), log (base 10), and sqrt — combined with addition, subtraction, multiplication, division and exponentiation, including chained expressions like sin(x²) or e^(2x).
Yes — every step is shown in order, naming the rule applied (Power, Product, Quotient, Chain, Sum/Difference, or exponential/logarithmic differentiation) before each substitution, right through to the simplified final derivative.
Type "2x^2" or "2*x^2" — both work. The calculator automatically inserts multiplication signs in common cases like a number directly followed by x, but writing it explicitly with * avoids any ambiguity in more complex expressions.
The Product Rule applies when two separate functions are multiplied together, like x²·sin(x). The Chain Rule applies when one function is nested inside another, like sin(x²) — you differentiate the "outer" function first, then multiply by the derivative of the "inner" part.
No — calculators and phones aren't allowed in most board and entrance exams for calculus questions. Use this tool for practice and homework, but make sure you can apply each differentiation rule by hand for exam day.