Derivative Calculator

Compute the derivative of any function. Supports power, product, chain, and quotient rules. Enter your expression below — see the notation guide on the right for input tips.

Calculus Differentiation

Inputs

Input examples

f(x)

Variable

Order

What is a Derivative?

The derivative f′(x) gives the instantaneous rate of change of f at every point — geometrically, it is the slope of the tangent line to the curve at x.

The second derivative f″(x) describes concavity: f″ > 0 means the curve bends upward; f″ < 0 means it bends downward. At a stationary point, f″ > 0 confirms a local minimum and f″ < 0 a local maximum.

f′(x) > 0f is increasing

f′(x) < 0f is decreasing

f′(x) = 0stationary point

f″(x) > 0local minimum

f″(x) < 0local maximum

Live Graph

Function graph

Enter a function with one variable
to see its graph here

f(x) — original

f′(x) — derivative

Result

Enter a function above and press Calculate

Step-by-Step Solution

The steps appear here after you calculate.

Common Rules

f(x)	f′(x)	f″(x)

Combination Rules

Product Rule

Use when differentiating two functions multiplied together. Differentiate each in turn while keeping the other fixed, then add the results.

Example:

Quotient Rule

Use when differentiating one function divided by another. A helpful mnemonic: "low d-high minus high d-low, over low squared go."

Example:

Chain Rule

Use when differentiating a function inside another function. Differentiate the outer function (leaving the inner unchanged), then multiply by the derivative of the inner function.

Example:

Sources & Verification

References

J. Stewart, Calculus: Early Transcendentals, 8th ed., Cengage, 2016 — Ch. 3 (Differentiation Rules: Sec. 3.2 The Product and Quotient Rules, Sec. 3.4 The Chain Rule).
M. Spivak, Calculus, 4th ed., Publish or Perish, 2008 — Ch. 9 (Derivatives) and Ch. 10 (Differentiation).