What is the tangent line at a point?

How it works

The tangent line calculator finds the line that touches y = f(x) at x = a:

y = f(a) + f′(a)(x − a)

It is the Taylor polynomial of f of order 1 about a. A computer algebra system (nerdamer, open source) finds it with exact numbers and writes it expanded, as mx + b; the algebra runs in the background after you start typing.

The page shows three things:

• Tangent line y = the expanded line, exactly (for example 6x − 9, or x cos(1) − cos(1) + sin(1) for sin(x) at 1).
• Slope f′(a): the slope m, as a decimal to 10 significant figures.
• Point f(a): the y value of the point of tangency (a, f(a)), as a decimal to 10 significant figures (left out in the case step 6 names).

Every answer is checked before it is shown.

1. 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.
2. 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 they do not, the page says the slopes of f just left and right of a differ. That happens at a corner, a jump or a vertical tangent, and also where h is too wide for the curve (sin(x) at 10⁶, x^(4/3) at 0): the page does not claim f has no derivative.
3. The derivative the algebra uses is compared with a numeric difference quotient at 20 points (to 1 part in a million), and the coefficients f(a) and f′(a) with the checked derivative at a.
4. The slope m of the line L, read as (L(10⁸) − L(−10⁸))/(2 · 10⁸), or as (L(1) − L(−1))/2 when that is past the largest computer number (e^x at 700), must agree with the average of the two quotients in step 2 to 10⁻⁵ × max(1, |m|), plus 10⁻¹⁵ × |f(a)|/h for rounding.
5. A slope below the smallest normal computer number, 2⁻¹⁰²² (about 2.2 × 10⁻³⁰⁸), or a slope of 0 when the line still has an x term, has lost its digits (e^x at −800): the page gives no answer. x e^(−x) at 700 has slope about −6.9 × 10⁻³⁰², which is shown.
6. Point f(a) is L(a) worked out exactly by the algebra, when that value agrees with f(a) in decimals to 10⁻¹² × (|f(a)| + |f′(a) a|). It is shown when the algebra's text is 0 (cos(2x) at π/4 gives 0, not the 6.1 × 10⁻¹⁷ that π rounded to a computer number gives), or when rounding a to a computer number cannot move its 10th figure: |f′(a) a| × 2.2 × 10⁻¹⁶ under 5 × 10⁻¹¹ × |f(a)|. Otherwise the point is left out (x³ − 2x at √2).

If any check fails, the algebra finds no formula, or the work takes over 3 seconds, the page says "No verified answer" (or why not) and shows no line.

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: 3, −1.5, pi/4, e.

How answers are written

• The line is written expanded, in the syntax you type: ^ for powers, a number or a space before a letter for multiplication, π for pi, ln for the natural logarithm. The terms can come in any order (1 − x/4 is the line −x/4 + 1), and like terms that hold π or e may stay apart (3x π^2 − 2 π^3 − 2x).
• Numbers in the line are exact: fractions such as 1/6, and values such as sqrt(3)/2, e^20 or cos(1). A line 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: the page says "No verified answer".
• The slope and the point are decimals to 10 significant figures, rounded half up.

What gets no answer

• f not defined at a, f(a) too large, or slopes just left and right of a that differ (|x| or the cube root of x at 0).
• 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. Where f is not a real number the curve has a gap.

Worked examples by hand

f(x) = x² at x = 3 (OpenStax Calculus Volume 1, section 3.1, Example 3.1). f′(x) = 2x, so the slope is m = f′(3) = 6. The point is (3, f(3)) = (3, 9). The line is y = 9 + 6(x − 3) = 6x − 9.

f(x) = 1/x at x = 2 (OpenStax Calculus Volume 1, section 3.1, Example 3.3). f′(x) = −1/x², so m = f′(2) = −1/4. The point is (2, 1/2). The line is y = 1/2 − (x − 2)/4 = 1 − x/4, that is y = −x/4 + 1.