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Chatty reference · maths1 week 8

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1 min read
2026-08-16

8. Maths Week 8 — Derivatives, Tangents & Critical Points

Officially:
  • Differentiability
  • Derivative
  • Computing derivatives
  • L'Hôpital's rule
  • Tangents
  • Linear approximation
  • Critical points
  • Local maxima/minima. citeturn1view0

Core object

f′(x)f'(x)
Interpretations:
  • instantaneous rate of change
  • slope of tangent
  • local behaviour

Patterns

Derivative computation

Apply appropriate rules.

Tangent

At x=ax=a:
y−f(a)=f′(a)(x−a)y-f(a)=f'(a)(x-a)

Linear approximation

f(x)≈f(a)+f′(a)(x−a)f(x)\approx f(a)+f'(a)(x-a)

Critical points

Typically solve:
f′(x)=0f'(x)=0
and also consider where f′f' is undefined.

Local extrema

Use derivative behaviour around critical points.

L'Hôpital

For appropriate indeterminate forms:
= \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ when the conditions for the rule hold. ---
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