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Derivative slope

Drag two points on a curve and shrink the gap h: the secant line turns into the tangent and its slope settles on f′(x). See the slope graph f′(x) drawn under f(x), a table of slopes from both sides, first-principles working for xⁿ and sin x, corners where there is no derivative, and fresh practice questions.

Keep it one tap away for everyday use.

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  1. Pick f(x)
  2. Drag P, Q
  3. Shrink h
  4. Slope f′
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Pick a function

Level

Secant to tangent

Drag along the curve to move P, or drag Q. Blue: the secant line PQ. Dashed: the tangent at P.

The slope graph f′(x)

Shrink h: the slope settles

hslope of PQ (h > 0)slope (h < 0)

From first principles

    Try this

    • With f(x) = x², put P at x = 1 and press Slide Q to P. Watch the slope of PQ go 3, 2.1, 2.01 … towards 2.
    • Tick Q on the left: the slopes come up from below (1.9, 1.99 …) to the same 2.
    • Pick |x| and put P at 0. From the right the slope is 1, from the left −1: no single tangent, so no derivative there.
    • Pick sin x and look at the slope graph: it is cos x.
    • Pick eˣ: the slope graph sits right on top of the curve.

    Check yourself

    Fresh numbers every time. Type a number or a fraction such as 3/4; nothing is saved or sent.

      What it shows

      The derivative of a function at a point is the slope of the curve there: how steeply it climbs or falls. This page lets you see that number appear. Two points sit on the curve: P at x = a and Q a small step h further along. The straight line through them is a secant, and its slope is

      slope of PQ = [f(a + h) − f(a)] / h.

      As you slide Q towards P (h → 0), the secant swings round and lands on the tangent at P, the dashed orange line. The number it settles on is the derivative f′(a). Under the main picture, the slope graph draws f′(x) for every x, so you can see that the slope of x² is 2x, the slope of sin x is cos x and the slope of eˣ is eˣ itself.

      It follows the school topics of limits and derivatives in Class 11 maths and continuity and differentiability in Class 12 maths. Before this, the idea of a limit is easiest to see with a graphing calculator; afterwards, the symbolic algebra tool differentiates any expression for you, and numerical methods teacher shows how computers estimate slopes.

      How to use it

      1. Pick a function from the list (x², x³ − 3x, sin x, cos x, eˣ, ln x, √x, |x|, 1/x), or choose My own… and type one, such as x^3/3 - x or sin(x)/x. Use ^ for powers, sqrt( ), abs( ), e^x and brackets after sin, cos and tan.
      2. Move P: drag along the picture or use the Point P slider.
      3. Shrink h: use the Gap h slider, drag Q itself, or press Slide Q to P to watch h fall from about 1 to 0.001.
      4. Tick Q on the left to make h negative and come at P from the other side.
      5. Read the formula line: it fills in your numbers, such as [f(2) − f(1)] / 1 = (4 − 1) / 1 = 3.

      The Level switch adds Class 12 detail: the left-hand and right-hand slopes and whether the function is differentiable at P.

      What each part means

      • Slope of PQ: the secant slope for the h you chose.
      • Tangent slope f′(a): the exact derivative for the ready-made functions (worked out numerically for a typed one).
      • Gap to the tangent: how far the secant slope still is from f′(a). It shrinks roughly in step with h.
      • Tangent line at P: y − f(a) = f′(a)(x − a), written as y = mx + c.
      • The table lists the secant slope for h = 1, 0.1, 0.01, 0.001 and 0.0001 from both sides. When both columns head for the same number, the derivative exists.
      • From first principles writes out the working for xⁿ (n from 1 to 6, using the binomial theorem) and for sin x (using the sum-to-product formula and the limit sin θ / θ → 1).

      Worked example

      Take f(x) = x² and put P at x = 1. With h = 1, Q is at x = 2 and the slope of PQ is (4 − 1) / 1 = 3. With h = 0.1 it is (1.21 − 1) / 0.1 = 2.1; with h = 0.01, 2.01; with h = 0.001, 2.001. From the left (h = −0.1) the slopes are 1.9, 1.99, 1.999. Both sides close in on 2, which is f′(1) = 2 × 1. The tangent at P is y = 2x − 1.

      Corners: when there is no derivative

      Choose |x| and put P at 0. From the right every secant has slope 1, from the left −1. The two sides never agree, so there is no single tangent: |x| is continuous at 0 but not differentiable there. Switch the Level to Class 12 to see the left-hand and right-hand slopes side by side.

      Check yourself

      At the bottom are five short questions with fresh numbers every time: a derivative of a quadratic, a secant slope, the power rule, a trig derivative and a corner. Type an answer (fractions like 3/4 are fine), press Check, and open the working if you want to see the steps. Nothing is saved.

      FAQ

      Why not just put h = 0? Then Q sits on P and the slope is 0/0, which means nothing. The derivative is what the slope approaches as h gets close to 0, which is why limits come first.

      Why does the slope graph of 1/x stop at 0? The function is not defined at x = 0, so it has no slope there either.

      Is this the same as my textbook? The ideas are the standard ones taught in school maths everywhere. The examples and pictures here are made for this page; it is an independent study aid.

      Where the numbers come from

      • Standard calculus: the derivative as the limit of [f(a + h) − f(a)] / h, the binomial theorem and sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2). Written for fibub; no textbook text, figures or exercises are used.
      • Checked against sympy 1.14: derivatives, second derivatives and antiderivatives of the ready-made functions at three points, the expansions of (x + h)ⁿ for n = 1 to 6, and a typed function's derivative.

      Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.

      Last updated 2026-11-09.

      Change log for Derivative slope

      1. Add #1181 Derivative as a slope Details on GitHub

      Every change to fibub

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