Derivative and Tangent Line Explorer

The derivative is a function, not an answer

Most tools that differentiate will hand you an expression and stop. That answers a question, but usually not the one people are stuck on, which is why the gradient of a curve should turn out to be a curve of its own.

The two panels here share a single x axis, and that alignment is the whole point. Pick any vertical line through the scene: the upper curve tells you the value of the function there, and the lower one tells you how steeply it is changing. Watch the tangent sweep and the lower curve is drawn as it goes, one gradient at a time, which is precisely how a derivative is assembled.

Where the gradient crosses zero

A flat tangent means a turning point. Because the panels are aligned, the zero crossing in the lower curve sits directly beneath the peak or trough that caused it. On the default cubic, x³ - 3x, that happens at x = -1 and x = 1, which is exactly where 3x² - 3 = 0.

The sign tells you the rest. Where the gradient curve sits above zero the function is rising; below zero it is falling. Try 1/x: its gradient is negative everywhere in view, which is the same statement as the curve falling everywhere, said twice.

Watching a limit actually converge

The derivative is defined as a limit, and a limit is a promise about what happens as something gets small. The step slider lets you keep that promise honest. The difference quotient

(f(x + h) - f(x)) / h

is not the derivative for any particular h; it is an approximation that improves as h shrinks. Set the step to 1 and the estimated curve is visibly the wrong shape. Bring it down to 0.01 and the two curves become hard to tell apart. Nothing is hidden in that transition, and it is a more convincing account of what a limit means than any amount of notation.

Two ways to take a difference, and why one is better

The forward difference above is the textbook definition. The central difference instead straddles the point:

(f(x + h) - f(x - h)) / 2h

Both evaluate the function twice, so they cost exactly the same. But the central version is symmetric about the point, which cancels the leading error term. In practice: halve the step and the central error drops to a quarter, while the forward error only halves. Switch the method with the step held fixed and the gap against the exact curve visibly changes.

There is a neat special case worth trying. Choose x² and the central difference is exact at every step, even h = 1. The error term it discards depends on the third derivative, and a quadratic’s third derivative is zero.

Where functions stop existing

Three of the functions here are undefined somewhere: ln x at zero and below, √x below zero, and 1/x at zero. The curves are cut off at those boundaries rather than flattened along them, because a flat segment would assert a value that does not exist.

A subtler case shows up in the readouts. Near the left edge of ln x, with a wide step, the exact gradient is still reported but the estimate is not. The derivative is perfectly well defined there; it is the finite difference that fails, because it needs f(x - h) and that point lies outside the domain. Distinguishing those two failures is the difference between a tool that is honest and one that quietly prints a meaningless number.

Your own function

Pick Type your own and the presets give way to a box. Whatever you type is differentiated symbolically, by the same rules you would apply on paper, so the exact gradient stays exact and the error column keeps meaning what it says. The derivative itself is shown in the hint under the gradient readout, so you can check it against your working rather than taking it on trust.

Some functions come back refused, with a reason. abs(x) has a corner at zero, floor(x) is flat between jumps, and min bends where its arguments cross. None of them has a gradient everywhere. The answer usually given for abs is sign(x), which is wrong at precisely the point anybody studying it is looking at, so this refuses instead of being confidently wrong there.

The viewing window is not a guess either. It is the widest stretch between -6 and 6 where the function was found to have values, which is why ln x opens on the positive side and 1/x does not straddle its pole.

Common mistakes

• Reading the lower curve as a second copy of the function. It is not the function at a different scale. It is the slope of the function, and its units differ.
• Thinking a large function value means a large gradient. Try √x: the function grows slowly and steadily, yet its gradient is unbounded at the origin. Height and steepness are independent.
• Believing a smaller step is always better. In exact arithmetic it is. In floating point, shrinking h far enough subtracts two nearly equal numbers and the answer degrades again. The slider stops at 0.001, comfortably above where that starts to bite.
• Expecting the tangent to touch at only one point. It is the line matching the curve’s slope at the marked point. On a cubic it will generally cross the curve again somewhere else, and that is fine.
• Assuming the gradient is zero wherever the curve looks flat. Zoom is deceptive. Read the number rather than the picture, which is why it is printed beside the marker.