$Derivative Calculator$

Derivative Calculator

Differentiate any function with the rules shown step by step, the derivative at a point, the tangent line, the second derivative and a graph.

What to find:

The derivative f′(x)

The derivative at x₀ and the tangent line

The second derivative f″(x)

The derivative of order n

f(x) =

Examples: x^3 − 2x, sin(2x), e^x, ln(x), sqrt(x), x^x, (x + 1)/(x − 1). The multiplication sign can be omitted: 2x, x sin x. The order n is from 1 to 6; x₀ can be entered as pi/4.

Test yourself

The derivative calculator differentiates a function with a step-by-step solution: it shows which rules apply (sum, product, quotient, chain rule), simplifies the result and lists the formulas used. It also finds the derivative at a point and the tangent line, the second derivative and derivatives up to order 6, and plots the function and its derivative.

Differentiation rules

Rule	Formula
Sum	(u ± v)′ = u′ ± v′
Constant multiple	(Cu)′ = Cu′
Product	(uv)′ = u′v + uv′
Quotient	(u/v)′ = (u′v − uv′)/v²
Chain rule	(f(u))′ = f′(u) · u′

Table of derivatives

(xⁿ)′ = nxⁿ⁻¹	(√x)′ = 1/(2√x)	(eˣ)′ = eˣ	(aˣ)′ = aˣ ln a
(ln x)′ = 1/x	(sin x)′ = cos x	(cos x)′ = −sin x	(tan x)′ = 1/cos²x
(cot x)′ = −1/sin²x	(arcsin x)′ = 1/√(1 − x²)	(arctan x)′ = 1/(1 + x²)	(C)′ = 0

Example 1. f(x) = x² sin x.
f′(x) = (x²)′ sin x + x² (sin x)′ = 2x sin x + x² cos x.

Example 2. f(x) = x³ − 3x, x₀ = 2.
f′(x) = 3x² − 3, f′(2) = 9, f(2) = 2, tangent line: y = 9x − 16.

Geometric and physical meaning

The derivative at a point is the slope of the tangent line: k = tan α = f′(x₀). The tangent line is y = f(x₀) + f′(x₀)(x − x₀). In physics the derivative of position with respect to time is the instantaneous velocity, and the derivative of velocity is the acceleration.