Interactive differentiation

Drag the point along the curve and see the tangent line move in real time. The derivative is the slope of the tangent.

The derivative:

f'(x₀) = lim[h→0] (f(x₀+h) - f(x₀)) / h

In practice we use differentiation rules. The derivative equals the slope of the tangent line at the point.

Function f(x)

x₀ = 0.0 — point on curve

f(x₀)

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f'(x₀) = slope

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Tangent line

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Derivative f'(x)

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Differentiation rules

(xⁿ)' = n·xⁿ⁻¹ Power rule

(c)' = 0 Constant

(eˣ)' = eˣ Exponential

(ln x)' = 1/x Natural log

(sin x)' = cos x Sine

(cos x)' = -sin x Cosine

(u·v)' = u'v + uv' Product rule

(u/v)' = (u'v-uv')/v² Quotient rule

Key insight: When f'(x₀) = 0, the tangent is horizontal — this is a maximum or minimum point!

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance.

— John von Neumann (1903–1957)