Derivatives & Tangents
Derivatives & Tangents is a free interactive simulation from StudyFi Lab. It runs right in your browser, with no sign-up and nothing to install.
Tangent, secant and the graph of f′ for your own function
Experiments
Tangent and derivative
Drag the point along the graph of f. The dashed line is the tangent: it touches the curve at x₀ and has the slope k = f′(x₀). The lower graph collects these slopes for every x – that is the derivative f′(x). Where f rises, f′ is positive; where it falls, f′ is negative.
Try this: Pick the cubic and move the point to where the tangent is horizontal. What value does f′ show there, and where does the lower graph cross the x-axis?
From secant to tangent
A secant passes through two points of the graph, x₀ and x₀ + h. Its slope is the difference quotient k(h) = (f(x₀ + h) − f(x₀)) / h. As h gets smaller, the second point slides towards the first and the secant turns into the tangent: the limit of k(h) for h → 0 is the derivative f′(x₀).
Try this: Let h shrink towards 0 (with the slider or the animation) and watch the secant slope in the values close in on f′(x₀). Try it with the sine as well.
Match the derivative
Above is a function f, below are three graphs, and only one of them is its derivative. Read f like a map of slopes: where f rises, f′ lies above the x-axis; where it falls, below; at a peak or in a valley, f′ passes through zero. Five rounds, a different kind of function each time.
Try this: Before you choose, find the points where f has a horizontal tangent and check which candidate is zero exactly there.
Maxima, minima and inflection
Stationary points are where f′(x) = 0 and the tangent is horizontal. If f′ changes from + to −, f has a local maximum there; from − to +, a local minimum; if the sign stays the same, it is a stationary inflection point. Where f″ changes sign, the curve switches between concave up and concave down: an inflection point.
Try this: Classify each “?” on the curve from the signs of f′ below, then switch on the classification to check. Finally set the cubic to a = 1, b = 0, c = 0.
Frequently asked questions
What is Derivatives & Tangents?
Derivatives & Tangents is a free interactive simulation from StudyFi Lab. It runs right in your browser, with no sign-up and nothing to install. Tangent, secant and the graph of f′ for your own function
Is Derivatives & Tangents free?
Yes. Derivatives & Tangents is completely free. It works in the browser on computers, tablets and phones, and you do not need an account.
What can I explore in Derivatives & Tangents?
Tangent and derivative, From secant to tangent, Match the derivative, Maxima, minima and inflection.
What does the “Tangent and derivative” experiment show?
Drag the point along the graph of f. The dashed line is the tangent: it touches the curve at x₀ and has the slope k = f′(x₀). The lower graph collects these slopes for every x – that is the derivative f′(x). Where f rises, f′ is positive; where it falls, f′ is negative.
What does the “From secant to tangent” experiment show?
A secant passes through two points of the graph, x₀ and x₀ + h. Its slope is the difference quotient k(h) = (f(x₀ + h) − f(x₀)) / h. As h gets smaller, the second point slides towards the first and the secant turns into the tangent: the limit of k(h) for h → 0 is the derivative f′(x₀).
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