Numerical derivative calculator

How to find the derivative of a function at a point

The derivative f′(x₀) is the slope of f at the point x₀: f′(x) = lim(h→0) [f(x+h)−f(x)]/h. For f(x) = x³, the power rule gives 3x², so the slope at x = 2 is 3 × 2² = 12, and the tangent line there is y = 12x − 16.

The answer here is numerical: the value of f′(x₀) or f″(x₀) at one point, to about 20 significant digits. For the formula itself, use the rules below or the table of common derivatives.

Derivative rules: power, product, quotient and chain

• Power rule. The derivative of xⁿ is nxⁿ⁻¹, so x³ gives 3x². A constant has derivative 0, and x has derivative 1.
• Product rule. (fg)′ = f′g + fg′. For x² sin x, that is 2x sin x + x² cos x, which at x = 1 equals 2 sin 1 + cos 1 ≈ 2.22324427548.
• Quotient rule. (f/g)′ = (f′g − fg′)/g², wherever g is not zero.
• Chain rule. Differentiate the outer function at the inner value, then multiply by the inner derivative: sin(2x) gives 2cos(2x).

The OpenStax differentiation rules work through the power, product and quotient rules with examples.

Common derivatives: trig, exponential and inverse functions

Trig arguments are in radians. The last column shows where each formula holds: tan, sec and cot need a nonzero denominator, and asin and acos have finite derivatives only for −1 < x < 1. atan(x) is the inverse tangent; cot(x) is 1/tan(x).

In xˣ the exponent varies too, so logarithmic differentiation gives xˣ(ln x + 1). Treating the exponent as a constant gives x · xˣ⁻¹ = xˣ, which misses the ln x term. At x = 2 the slope is 4(ln 2 + 1) ≈ 6.7726.

ln(sec x) has a real logarithm only where sec x > 0. √x has derivative 1/(2√x) for x > 0 and no finite derivative at 0. The OpenStax exponential and logarithm lesson derives these rules.

Derivative values at a point

Order 1 is the slope f′; order 2 is f″, how fast the slope itself changes. For ln x, f″(x) = −1/x², so f″(2) = −0.25. A second derivative is not the square of the first.

The tangent line at x₀ is y = f(x₀)+f′(x₀)(x−x₀). For x³ at 2, f(2) = 8 and f′(2) = 12, so y = 8 + 12(x − 2) = 12x − 16. A second derivative gives no tangent line, because it is not a slope.

How to type the function

• Use x as the only variable; 2x and 2*x both work.
• Trig functions use radians, so sin(x) at x = 30 means 30 radians.
• Write exponent signs in full. A search for "e 2x" can mean exp(2*x), with derivative 2e^(2x), or exp(-2*x), with derivative −2e^(−2x).
• Bracket the whole argument: sin(2*x) and 2*sin(x) are different functions with different derivatives.
• Type powers and roots as x^2, sqrt(x) or x^(1/2).

How the numerical derivative is calculated

The first derivative starts from the central difference quotient [f(x₀ + h) − f(x₀ − h)]/2h, with h = 0.1 × max(1, |x₀|) halved repeatedly. Richardson extrapolation (Ridders' method, Numerical Recipes §5.7) combines the quotients.

Each extrapolation step cancels one more of the h², h⁴, … error terms, which takes the result to about 20 significant digits.

The second derivative uses [f(x₀ + h) − 2f(x₀) + f(x₀ − h)]/h² in the same way. The estimated error is the gap between the last two extrapolated estimates: a measure of how well they settled, not a guaranteed bound.

Slopes from the left and right are compared too. abs(x) at 0 has slope −1 from the left and +1 from the right, so it has no derivative there, even though the central quotient averages to 0. 1/x and √x have no derivative at 0 either.

Common mistakes

• Dropping the chain-rule factor. The derivative of sin(2x) is 2cos(2x); writing cos(2x) gives half the slope.
• Working in degrees. The derivative of sin x is cos x only when x is in radians.
• Using the power rule on xˣ. It applies only when the exponent is a constant.
• Checking a formula at one point only. Matching values at a few points is evidence that a derivative formula is right, not proof.

A derivative is a rate at one point. To add up change over an interval, use the definite integral calculator.