Step-by-step derivatives with every rule named: product, quotient and chain rule, a simplified f′(x), the second derivative, the value at any point and the tangent line in y = mx + b form.
Derivative Calculator - Step by Step, Tangent Line, f″(x)
Step-by-step derivatives with every rule named: product, quotient and chain rule, a simplified f′(x), the second derivative, the value at any point and the tangent line in y = mx + b form.
Parameters
Enter data for calculations
💡 Fill in all required fields to unlock the calculate button
An atlas of derivatives, worked out for your function
The derivative of x2sin(x) is 2x·sin(x) + x2·cos(x), and at x = π/2 its value is 3.141592654, so the tangent line there is y = 3.141592654x - 2.4674011. This derivative calculator differentiates symbolically, the way you would on paper: it names the rule used at every stage (product, quotient, chain, power), simplifies the answer, and, when you give a point, returns f(x₀), f′(x₀), the tangent line in slope-intercept form and, on request, the second derivative.
Three boxes and the order to read the answer
- Function f(x) - a formula in x, such as 3x^4 - 2x + 5, x^2 sin(x) or ln(x^2+1). Use ^ for powers; the multiplication sign can be left out (2x, x sin(x)). Write sin²x as sin^2(x) and put a longer exponent in parentheses: e^(3x).
- Point x₀ (optional) - a number or an expression without x, for example 2, -0.5 or pi/4. Use a period for decimals. Leave it empty and you get only the formula of the derivative.
- Order of the derivative - first only, or first and second. The second derivative usually settles whether a point with f′(x₀) = 0 is a minimum or a maximum.
- Read the result - the dark box holds f′(x) (and f″(x)), the tiles below hold the values at x₀, then the tangent line, the function as the calculator read it, and the list of rules applied, from the outermost operation inward.
Accepted names: sin, cos, tan, cot, sec, csc, ln, log, exp, sqrt, abs, arcsin, arccos, arctan, arccot, the constants pi and e, and x. Here log means the base-10 logarithm and ln the natural one, as on most US calculators. Angles are in radians.
What a derivative measures
The derivative of f at a point x₀ is the limit of the difference quotient [f(x₀ + h) - f(x₀)] / h as h shrinks to zero. Read on a graph, it is the slope of the tangent line at that point. Read as a rate, it is how fast f changes per unit of x right there: if f is position in feet and x is time in seconds, f′ is the velocity in feet per second.
Nobody evaluates that limit by hand for every new function. A short table of standard derivatives plus a handful of rules takes any formula apart: a sum splits into terms, a product goes to the product rule, a fraction to the quotient rule, a function inside another function to the chain rule. The calculator follows exactly that path and prints each move. For f(x) = x3 - 3x at x₀ = 1 the derivative is 0, so the tangent is the horizontal line y = -2.
The second derivative is the derivative of the derivative. It tells how the slope itself changes: positive means the graph bends up like a cup (concave up), negative means it bends down (concave down). For 3x4 - 2x + 5 the first derivative is 12x3 - 2 and the second is 36x2; at x₀ = 1 they are worth 10 and 36.
Table of derivatives: eighteen building blocks
Every answer the calculator gives is assembled from these entries. The last column is where the formula holds; outside it the function or its derivative is not defined.
| f(x) | f′(x) | Valid for |
|---|---|---|
| c (a constant) | 0 | all x |
| xn | n·xn-1 | depends on n |
| √x | 1/(2√x) | x > 0 |
| 1/x | -1/x2 | x ≠ 0 |
| ex | ex | all x |
| ax | ax·ln a | a > 0 |
| ln x | 1/x | x > 0 |
| log x (base 10) | 1/(x·ln 10) | x > 0 |
| sin x | cos x | all x |
| cos x | -sin x | all x |
| tan x | 1/cos2x = sec2x | cos x ≠ 0 |
| cot x | -1/sin2x = -csc2x | sin x ≠ 0 |
| sec x | sec x·tan x | cos x ≠ 0 |
| csc x | -csc x·cot x | sin x ≠ 0 |
| arcsin x | 1/√(1 - x2) | -1 < x < 1 |
| arccos x | -1/√(1 - x2) | -1 < x < 1 |
| arctan x | 1/(1 + x2) | all x |
| arccot x | -1/(1 + x2) | all x |
The rules, sorted by the shape of the formula
The step list in the result uses these names. Which rule fires depends only on the outermost operation of the expression; the calculator then works inward. Every example in the right column is the calculator's own output, simplification included.
| Rule | Formula | Example from the calculator |
|---|---|---|
| Sum and difference | (u ± v)′ = u′ ± v′ | (3x4 - 2x + 5)′ = 12x3 - 2 |
| Constant multiple | (c·u)′ = c·u′ | (πx2)′ = 2π·x |
| Power | (un)′ = n·un-1·u′ | ((x2 + 1)3)′ = 6(x2 + 1)2·x |
| Product | (u·v)′ = u′·v + u·v′ | (x2sin x)′ = 2x·sin(x) + x2·cos(x) |
| Quotient | (u/v)′ = (u′·v - u·v′)/v2 | (x/(x2 + 1))′ = (1 - x2)/(x2 + 1)2 |
| Chain (composite function) | f(g(x))′ = f′(g(x))·g′(x) | (sin(x2))′ = 2x·cos(x2) |
| Exponential | (au)′ = au·ln(a)·u′ | (2x)′ = 2x·ln(2) |
| x in base and exponent | (uv)′ = uv·(v′·ln u + v·u′/u) | (xx)′ = xx·(ln(x) + 1) |
Six functions run through the calculator
f(2) = 5, f′(2) = -1, f″(2) = 4.
Tangent: y = -x + 7.
f′(x) = 3e3x, f′(0) = 3.
Tangent: y = 3x + 1.
f(3) = 5, f′(x) = x/√(x2 + 16), f′(3) = 0.6.
Tangent: y = 0.6x + 3.2.
f′(x) = 3x2 - 12x + 9, f′(3) = 0, f″(3) = 6: local minimum, tangent y = 1.
f′(x) = 2/(1 + 4x2), f′(0.5) = 1.
Tangent: y = x + 0.2853981634.
f(2) = 4, f′(x) = xx·(ln(x) + 1), f′(2) = 6.772588722.
Tangent: y = 6.772588722x - 9.545177444.
Four derivatives that behave unexpectedly
ex is its own derivative. Up to a constant factor it is the only function with that property, which is why ekx turns up wherever the rate of change is proportional to the amount itself: continuous compounding, radioactive decay, unchecked population growth. For e3x the derivative is simply 3e3x.
A continuous function can have no derivative. |x| has no gap at 0, yet the difference quotient from the left is -1 and from the right 1. The graph has a corner, and the calculator reports both one-sided values. By contrast |x2| does have a derivative at 0 (it is 0), even though the formula for it gives 0/0 there; the calculator then takes the limit of the difference quotient and says so.
A vertical tangent is also a missing derivative. The cube root x1/3 at 0 has a tangent line that coincides with the y-axis. The difference quotient grows without bound as h shrinks, so there is no derivative even though the curve looks smooth.
A zero derivative is not always an extremum. x3 at 0 has f′ = 0 and f″ = 0, and the function keeps increasing straight through. That point is an inflection point with a horizontal tangent, which is why the calculator writes that the second derivative does not decide when f″(x₀) = 0.
Reading the signs of f′ and f″ at a point
This is the classification the result text uses after the tangent line. It is the first and second derivative test from a calculus course, applied to the single point you typed.
| At x₀ | What it means |
|---|---|
| f′(x₀) > 0 | f is increasing near x₀ |
| f′(x₀) < 0 | f is decreasing near x₀ |
| f′(x₀) = 0 and f″(x₀) > 0 | local minimum |
| f′(x₀) = 0 and f″(x₀) < 0 | local maximum |
| f′(x₀) = 0 and f″(x₀) = 0 | inconclusive: check the sign of f′ on both sides |
| f′(x₀) does not exist | a corner, a vertical tangent, or no function on one side |
The first two rows assume the derivative is continuous at x₀, which holds for functions built from the table above except at points where the formula for f′ stops making sense. The verdict is about a neighborhood of the point, not about the whole graph: a local minimum can sit well above the lowest value of the function elsewhere.
Derivative questions from homework and review sessions
Related tools
System of Equations Calculator
Two or three linear equations solved step by step, with exact fractions and a check - See the calculator
Definite Integral Calculator
Value of a definite integral with error estimate, area, average value and Simpson comparison - See the calculator
Linear Function Calculator
Slope, intercepts and the equation of a line, the natural next step after a tangent line - See the calculator
Quadratic Function Calculator
Vertex, roots and graph data of a parabola, the place where its derivative equals zero - See the calculator
Logarithm Calculator
Logarithms in any base, handy when a derivative leaves ln a or ln 10 in the answer - See the calculator
Powers and Roots Calculator
Exact powers and roots for checking values such as e2 or the cube root at a point - See the calculator
Series Convergence Calculator
Convergence tests for infinite series, the next chapter in most calculus courses - See the calculator
Laplace Transform Calculator
Transform pairs that turn derivatives into multiplication by s, used for differential equations - See the calculator