differentiation rules and techniques — HSC mathematics advanced, live

Differentiation Rules and Techniques

The power rule states that for f(x) = x^n, the derivative is f'(x) = nx^(n-1), which applies to polynomials and rational functions. The chain rule, product rule, and quotient rule extend our capacity to differentiate composite functions, products of functions, and quotients of functions respectively. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

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why this one carries the topic.

Calculus is one machine run in two directions: differentiation takes a function and hands you its rate of change; integration takes the rate and hands the function back. The Fundamental Theorem is the proof they're inverses — which is why every substitution is just differentiation read backwards.

The power rule gives nx^(n-1) for basic polynomials, while chain, product and quotient rules handle composite, multiplied and divided functions respectively.

what examiners catch — Examiners test whether students correctly apply the chain rule to nested functions like (3x² + 1)⁵ by multiplying the outer derivative by the inner derivative, not just differentiating the outer function alone.

three things, not forty.

Chain rule: if y = f(g(x)), then dy/dx = f'(g(x)) × g'(x), essential for differentiating composite functions like (3x² + 1)⁵

Product rule: if y = u(x)v(x), then dy/dx = u'v + uv', used for functions like x²sin(x)

Quotient rule: if y = u(x)/v(x), then dy/dx = (u'v - uv')/v², applicable to rational functions like (2x + 1)/(x² - 3)

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knowhere maps every concept to what it rests on and what rests on it — 1 underneath this one, 4 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptdifferentiation rules and techniquesThe power rule gives nx^(n-1) for basic polynomials, while chain, product and quotient rules handle composite, multiplied and divided functions respectively.

sits under itthe derivative as an instantaneous rate of changebecause the rules are shortcuts to a thing you must first understand

built on itimplicit differentiation and related ratesbecause implicit differentiation is the chain rule applied to every y-term

built on itthe chain, product and quotient rulesbecause the advanced rules extend the basic kit

built on itstationary points and their naturebecause finding f′=0 requires finding f′

built on ittangent lines and linear approximationbecause the tangent's slope is the derivative, evaluated

3 more, same treatment.

Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.

antiderivatives and indefinite integrationlive →definite integration and the fundamental theorem of calculusin the appintegration by recognition and substitutionin the app

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