Derivative & tangent line
The derivative isn't just a formula — it's the slope of the line that just grazes the curve at a single point. So drag the point and watch. Pick a curve, slide x₀ along it, and the tangent line — and its slope f'(x₀) — update live, in real time.
Watch the curve straighten into its tangent line as you zoom in — this is what “the derivative is a local slope” really means.
The ideas you're seeing
What the derivative IS
f'(x₀) is defined as the limiting slope of a secant line as two points on the curve get infinitely close together — geometrically, it's exactly the slope of the tangent line at that single point.
The tangent line equation
Once you know f(x₀) and f'(x₀), the tangent is just a straight line through (x₀, f(x₀)) with slope f'(x₀): y = f(x₀) + f'(x₀)(x − x₀) — the same point-slope form from coordinate geometry.
Where the slope is zero
Points where f'(x₀) = 0 have a horizontal tangent line — these are exactly the candidates for local maxima and minima, the core idea behind the next chapter, Application of Derivatives.
Derivative of sin(x)
d/dx(sin x) = cos x: notice the tangent's slope oscillates between −1 and +1, exactly matching where cos(x) is at each x₀. Try sliding through the sine curve and watch the slope trace out a cosine-shaped pattern.
Part of the Continuity and Differentiability chapter — read the notes, grab the formula sheet and take the quiz. One of Priodemy for School, free with every EduSuite school.
The derivative is a slope you can see
From average to instantaneous
The slope between two points on a curve is easy: rise over run. The difficulty is defining the slope at a single point, where there is no run to divide by. Calculus solves this with a limit. Take a second point a distance h away, compute the slope of the line joining them, then let h shrink towards zero. That is the definition f′(x) = limh→0 [f(x+h) − f(x)] / h.
Geometrically, the joining line — the secant — pivots as the second point slides closer, and in the limit it becomes the tangent. The derivative is the gradient of that tangent. Drag along the curve here and the tangent follows; the number reported is simply how steeply the curve is rising or falling at that exact point.
What the sign and size tell you
A positive derivative means the function is increasing there, negative means decreasing, and the magnitude says how steeply. This is why derivatives answer questions that have nothing to do with graphs: if the function is position against time, the derivative is velocity; if it is cost against quantity, the derivative is marginal cost. The tangent slope is the rate of change, whatever the variables happen to represent.
Where the tangent is horizontal the derivative is zero, and these stationary points are the candidates for maxima and minima. To classify one, check whether the derivative changes sign around it, or use the second derivative: negative indicates a maximum, positive a minimum. Watch the tangent flatten as you cross a peak and then tilt the other way — the sign change is the test, made visible.
Not every point has a derivative
Differentiability requires the limit to exist and agree from both sides, which is stricter than continuity. The function |x| is continuous at zero but has no derivative there: approach from the left and the slope is −1, from the right it is +1. Geometrically there is a sharp corner, and no single tangent line fits. Every differentiable function is continuous, but the reverse does not hold — a favourite exam point.
Mistakes that cost marks
Confusing f′(x) with f(x). The derivative being zero does not mean the function is zero; it means the graph is momentarily flat. A curve can sit far above the axis and still have a stationary point.
Assuming every stationary point is a maximum or minimum. A point of inflection such as x = 0 on y = x³ has zero derivative but is neither. You must test the sign change rather than stopping at f′(x) = 0.
Differentiating trigonometric functions in degrees. The result that the derivative of sin x is cos x holds only when x is in radians. In degrees an extra factor of π/180 appears, and forgetting this quietly invalidates the whole answer.