Calculus
Step-by-Step Derivative Calculator
Compute derivatives of any function instantly. Calculate derivatives using the power rule, chain rule, product rule, and quotient rule with complete step-by-step differentiation analysis.
d/dx
Examples:
Differentiation Rules
Power Rule
d/dx [x^n] = nx^(n-1)
Product Rule
[fg]' = f'g + fg'
Chain Rule
d/dx [f(g(x))] = f'(g(x))g'(x)
Constant Rule
d/dx [c] = 0
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Solution Methodology
01
Input Function
Enter the symbolic expression f(x) — supports polynomials, trig, exponentials, and logarithms.
02
Parse Syntax Tree
The engine tokenises the input and builds an abstract syntax tree to identify all nested operations.
03
Identify Applicable Rules
Each node is tagged: power rule, product rule, quotient rule, or chain rule as appropriate.
04
Differentiate Term by Term
Rules are applied recursively from innermost to outermost, with every step recorded.
05
Simplify & Display
The resulting derivative is algebraically simplified and each intermediate step is shown in sequence.
Common Questions
How do you calculate the derivative of a function?
A derivative represents the instantaneous rate of change of a function at any given point. It is formally defined as the limit f′(x) = lim(h→0) [f(x+h) − f(x)] / h. In practice, this limit is evaluated symbolically using differentiation rules — most commonly the power rule d/dx xⁿ = nxⁿ⁻¹ — so you rarely need to compute the limit directly. The result gives the slope of the tangent line to the curve at every point x.
Can this calculator handle the power rule and chain rule?
Yes. The engine automatically detects the structure of your expression and applies the power rule, chain rule, product rule, and quotient rule in the correct order. For example, differentiating sin(x³) triggers the chain rule: the outer function is sin(u) with derivative cos(u), and the inner function u = x³ has derivative 3x². The final result 3x²cos(x³) is assembled and simplified automatically.
What is the chain rule and when is it used?
The chain rule applies when differentiating a composite function — one function nested inside another. It states that d/dx f(g(x)) = f′(g(x)) · g′(x). Whenever the engine detects a function of a function (e.g. e^(x²), ln(sin x), or (3x+1)⁵) it automatically activates chain-rule mode, differentiates the outer shell first, substitutes the inner expression back, then multiplies by the derivative of the inner expression.
How do I verify my derivative answer step by step?
Enter your expression into the input field and the solver will display each differentiation step in a numbered list: rule identification, intermediate transformation, simplification, and final result. You can compare each intermediate line against your handwritten working to pinpoint exactly where a sign error or missed term occurred. The output also shows the simplified form alongside the unsimplified intermediate for full transparency.