What is the derivative of a function?

How it works

The derivative calculator finds f′, the derivative of the function f you type, with respect to its variable. A computer algebra system (nerdamer, open source) applies the differentiation rules. The algebra runs after you start typing, in the background.

Every answer is checked before it is shown. At 20 points (fixed pseudo-random numbers, mostly between −4 and 4, some up to ±10), the slope of f is measured numerically with a five-point central difference, step h = 0.0001 × max(1, |x|), and compared with the derivative formula; they must agree to 1 part in a million. Points where f is not a real number are skipped, and so are points next to a corner or a jump, where the steps h and h/2 disagree. An answer that fails is never shown; the page says "No verified answer" instead.

Then the answer is put in a textbook form:

• a derivative that is a polynomial is expanded: (x² + 2)(3x³ − 5x) gives 15x⁴ + 3x² − 10;
• a derivative that is a rational function (a ratio of polynomials) is written as one simplified fraction when the algebra finds one: 5x²/(4x + 3) gives 10x (2x + 3)/(4x + 3)²;
• anything else stays as the algebra gave it.

Each rewrite is its own checked step (equal to the first form at 20 points); if it fails, the first form is shown.

Value at a point. When you type a number a, the page evaluates the derivative formula at x = a. It shows the value only when it is a real number and a central difference of f at a, with h = 0.00001 × max(1, |a|), agrees with it to 1 part in 100,000. So a corner (|x| at 0), a vertical tangent (√x at 0) or a point outside the domain gives no value.

What you can type

• One variable. Use one lowercase letter as the variable: x, t, y, and so on; the derivative is with respect to that letter. With no letter the function is a constant and its derivative is 0. Two different letters give no answer. The letter e is Euler’s number, and pi (or π) is π.
• Numbers can have decimals (2.5) and powers of ten (1e-3).
• Operations: + − * / and ^ for powers. Brackets group. A number or bracket next to a letter multiplies: 2x, (x + 1)(x − 1), x sin(x).
• Functions: sqrt, cbrt, ln (and log, the same natural logarithm), log10, exp, abs (or |x|), sin, cos, tan, sec, csc, cot, asin, acos, atan (arcsin, arccos, arctan also work), sinh, cosh, tanh and their inverses. Angles are in radians. sin x means sin(x); sin x^2 means sin(x²).
• The point a is a number or a constant expression: 1, −2.5, pi/4, e.

How answers are written

The derivative uses the syntax you type: ^ for powers, sqrt(u), |u| for absolute value, π for pi, ln for the natural logarithm, a space or a number before a letter for multiplication. Terms go by power of the variable, highest first, and factors of a product lowest degree first. The page shows it as MathML; Copy answer gives this text. An answer holding a number the algebra rounded (a fraction whose denominator, after removing factors 2 and 5, is over 1,000,000, or a decimal with more than 12 significant digits) is not shown. The value at a is a decimal to 10 significant figures.

Assumptions

• Angles are in radians; ln and log are the natural logarithm.
• The derivative formula may be defined at points where f is not, or the other way round; the value at a follows the rule above.
• An answer that fails its check, finds no formula, or takes over 3 seconds is not shown ("No verified answer").

Worked examples by hand

j(x) = (x² + 2)(3x³ − 5x) (OpenStax Calculus Volume 1, section 3.3, Example 3.24). By the product rule, j′(x) = 2x(3x³ − 5x) + (x² + 2)(9x² − 5) = 6x⁴ − 10x² + 9x⁴ − 5x² + 18x² − 10 = 15x⁴ + 3x² − 10.

f(x) = x² − 4x + 6 at x = 1 (OpenStax Calculus Volume 1, section 3.3, Example 3.22). f′(x) = 2x − 4, so f′(1) = −2. With f(1) = 3, the tangent line is y = 3 − 2(x − 1) = −2x + 5.

f(x) = 6/x² (OpenStax Calculus Volume 1, section 3.3, Example 3.27). Write f(x) = 6x⁻²; the power rule gives f′(x) = −12x⁻³ = −12/x³.