More Derivative Examples#

Stage 3 · 📉 Derivatives & the Computation Graph · Lesson 10 of 17 · intermediate

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Important

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When the slope changes#

The line \(f(a) = 3a\) had the same slope everywhere. Most functions do not — their derivative changes from point to point. This lesson makes that concrete, because a neuron’s sigmoid is exactly such a curve.

A curved example#

Take \(f(a) = a^2\). At \(a = 2\), \(f = 4\); nudge to \(a = 2.001\) and \(f \approx 4.004\) — a slope of about 4. But at \(a = 5\), \(f = 25\); nudge to \(5.001\) and \(f \approx 25.010\) — a slope of about 10. The slope is twice the input, which is exactly the rule

\[f(a) = a^2 \;\Rightarrow\; \frac{df}{da} = 2a.\]

The derivative is now a function of \(a\), not a constant.

A few more#

The same pattern holds across the standard functions — \(f(a) = a^3\) has derivative \(3a^2\), and \(f(a) = \ln a\) has derivative \(1/a\). You need not re-derive these from nudges each time; they are tabulated in any calculus reference. What matters is reading them the same way: how fast does the output move as I wiggle the input, right here?

The takeaway#

For a curve, “the derivative” always means the slope at a particular point. That single idea — a slope that varies — is all the calculus the rest of the course needs. Next we organise a multi-step computation so these per-point slopes can be combined mechanically, through a computation graph.

Hint