Find the instantaneous rate of change

How it works

The instantaneous rate of change of f at x = a is the derivative there:

f′(a) = lim (f(a + h) − f(a))/h as h → 0

A computer algebra system (nerdamer, open source) finds it with exact numbers: it finds the tangent line L(x) = f(a) + f′(a)(x − a) (the Taylor polynomial of order 1 about a, expanded), and the exact rate is the derivative of L, or when that fails its check, L(1) − L(0) simplified. It also finds the derivative f′(x) of f. The algebra runs in the background after you start typing.

The page shows:

• Rate of change f′(a): exactly (for example 8, −8/5, cos(1), sqrt(e)); left out when neither route gives a checked exact form.
• As a decimal: the exact rate to 10 significant figures, rounded half up, or (L(10⁸) − L(−10⁸))/(2 · 10⁸) when there is no exact form ((L(1) − L(−1))/2 when that is past the largest computer number).
• Derivative f′(x): the derivative at every x, as the algebra writes it.

Every answer is checked before it is shown. f must be a real number at a: f(a) that is not a number, or is ±∞ while f is a number just beside a (at a ± 10⁻⁶ × max(1, |a|): ln(x) or 1/x at 0), means f is not defined at x = a; ±∞ on both sides as well means f(a) is past the largest computer number, about 1.8 × 10³⁰⁸ (e^x at 710), and the page says f(a) is too large. The one-sided difference quotients (f(a + h) − f(a))/h and (f(a) − f(a − h))/h, with h = 10⁻⁶ × max(1, |a|), must agree to 1 part in 1,000 of max(1, |right quotient|), plus 10⁻¹⁵ × |f(a)|/h for rounding; if not, the page says the slopes of f just left and right of a differ (a corner, a jump, a vertical tangent, or a curve that bends too sharply for h, such as sin(1/x) at 10⁻³). Each derivative the algebra finds is compared with a numeric difference quotient at 20 points (to 1 part in a million), and f(a) and f′(a) with the checked derivative at a. The rate must agree with the average of the two quotients to 10⁻⁵ × max(1, |f′(a)|), plus 10⁻¹⁵ × |f(a)|/h for rounding. If any check fails, the algebra finds no formula, or the work takes over 3 seconds, the page shows no answer.

What you can type

• A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
• The function f uses the variable x. 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, 3(x + 1), x sin(x).
• Constants: pi (or π) and e.
• 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 without brackets means sin(x); sin x^2 means sin(x²).
• The point a is a number or a constant expression: 2, −1.5, pi/2, e. Decimals in f become exact fractions: 0.4 is 2/5.

How answers are written

• The exact rate and f′(x) are written in the syntax you type, with exact numbers. A result that would hold a number the algebra could only give rounded (a whole number past 2⁵³; a fraction p/q, p and q being the numbers that multiply its top and its bottom, such as 288557167/(342919925e), when q after removing its factors 2 and 5 is over 1,000,000, or is over 1 while |p| times it is over 10¹²; or a decimal with more than 12 significant digits) is not shown.

What gets no answer

• f not defined at a, f(a) too large, or slopes just left and right of a that differ (a corner, jump or vertical tangent).
• A rate below 2⁻¹⁰²² (about 2.2 × 10⁻³⁰⁸, the smallest normal computer number), or 0 while the tangent line still has an x term (e^x at −800): it is too small for a computer number.
• A step that fails its check, finds no formula, or takes over 3 seconds.

The chart

The chart draws y = f(x) from x = a − 3 to a + 3, with the point (a, f(a)) marked.

Worked examples by hand

f(x) = 3x² − 4x + 1 at x = 2 (OpenStax Calculus Volume 1, section 3.1, Example 3.5). f′(x) = 6x − 4, so f′(2) = 12 − 4 = 8.

T(t) = 0.4t² − 4t + 70 at t = 3 (OpenStax Calculus Volume 1, section 3.1, Example 3.9: a temperature in °F, t in hours). Typed as 0.4x^2 − 4x + 70: f′(x) = 0.8x − 4, so f′(3) = 2.4 − 4 = −1.6 = −8/5: the temperature falls at 1.6 °F an hour.