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Calculus · Derivative function

Derivative function

The derivative function f′ gives, at each x, the slope of the tangent line to the graph of f, wherever that slope exists. Its graph sits below f: on an interval where f′ is positive, f increases; where it is negative, f decreases; at a peak or valley where f is differentiable, f′ is 0.

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Derivative function in this visualization

The derivative function of ff, written f′(x)f'(x) (also written y′y' or dydx\frac{dy}{dx}), assigns to each xx the derivative of ff there: the slope of the tangent line to the graph at xx. Where that slope does not exist, f′f' has no value. This page draws ff and f′f' one above the other on the same xx-scale, for eight functions.

Derivative function
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Also written f′(x0)=lim⁡Δx→0ΔyΔxf'(x_0) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}, the derivative at a point x0x_0; letting x0x_0 vary gives the derivative function, and this page's aa is x0x_0. The blue point P sits on ff at x=ax = a, with its tangent line. The small triangle under the tangent has a run of 1, so its rise is the tangent’s slope. The same signed number is drawn, on the lower graph’s own scale, as the yellow bar ending at the point (a,f′(a))(a, f'(a)). As aa moves, that point runs along the graph of f′; Trace draws it from left to right.

The two rows between the graphs are a sign chart: the upper row says whether ff rises (↗) or falls (↘), the lower one gives the sign of f′f' (+ or −, with 0 where f′=0f' = 0 and DNE where f′f' does not exist).

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Things to notice

  • On an interval where f′>0f' > 0, ff increases; where f′<0f' < 0, it decreases. The converse is weaker: x3x^3 increases everywhere, yet f′(0)=0f'(0) = 0. A peak or valley inside the interval at which ff is differentiable has f′=0f' = 0; but f′=0f' = 0 does not make one (again x3x^3 at 0), and a peak or valley can sit where f′f' does not exist, as at the corner of ∣x∣|x|.
  • Where the bending of ff changes direction (an inflection point) and f′′f'' changes sign, f′f' has a local maximum or minimum. On sin⁡x\sin x the inflection points sit over the peaks and valleys of cos⁡x\cos x, where sin⁡x\sin x is steepest. That is not true of every inflection point: x3\sqrt[3]{x} has one at 0, where its tangent is vertical and f′f' has no value.
  • The lower graph shows slopes, not values of ff: its height at xx says how fast ff changes there. The two graphs have their own vertical scales, chosen so that each fits.
  • At the corner of ∣x∣|x|, f′f' jumps from −1-1 to 11 and f′(0)f'(0) does not exist. At the cusp of x2/3x^{2/3}, f′f' tends to −∞-\infty from the left and +∞+\infty from the right; at the vertical tangent of x3\sqrt[3]{x}, to +∞+\infty from both sides. Each of these functions is continuous at 0, yet not differentiable there.
  • exe^x is its own derivative: the lower graph repeats the upper one.
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Controls

  • Drag P, the point on f′f', or anywhere on either graph to move aa; or turn the wheel under Point and press the key points. Trace runs P from the left end to the right end and draws f′f' behind it, stopping briefly at each peak, valley, inflection point and point without f′f' as its name lights up in the sign chart; P stays at the right end. Extrema & inflections draws dashed guides through the peaks, valleys and inflection points of ff, down through the graph of f′f', and names them in the sign chart.
  • Presentation mode opens a full-screen view for projectors (Space traces, ← → move aa, H hides the graph of f′f'; it then appears only where P has been).
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Related

Further reading: Wikipedia: Derivative; OpenStax, Calculus Volume 1, 3.2 The Derivative as a Function.