Differential calculus
Chapter 40 of 9 strong
RESTS ONUse principal inverse circular values,domains,ranges,transformations and equationsFOUNDATIONSIdentify the inner and outer functions of a composite, differentiate each, multiply the outer derivative evaluated at the inner function by the inner derivative, and write the result in the requested factorised formFOUNDATIONSConstruct direct, contrapositive and contradiction arguments with justified stepsCH 1Translate between a function graph's features and its gradient function: turning points become x-intercepts of f', positive and negative slopes become signs of f', polynomial degree drops by one, and points where the graph is not smooth or continuous have no derivativeFOUNDATIONSSimplify a polynomial to an explicit sum of powers where necessary and produce its complete derivative in descending-power form, including zero coefficients where the response table labels themFOUNDATIONSIdentify two factors, differentiate each, combine them as u'v + uv' in the stated order, and simplify or refactor the result only where the question asks for a factorised formFOUNDATIONSIdentify the numerator and denominator, apply (v u' − u v')/v², preserve the squared denominator, and state the non-zero denominator conditionFOUNDATIONSSubstitute an exact value into a derivative and compute the exact rational gradient, distinguishing f'(a) from f(a) and from the unsimplified derivative ruleFOUNDATIONSExtend the power rule to a rational exponent on the function's domain, including negative and fractional powers, and rewrite simple root and reciprocal forms as powers before differentiatingFOUNDATIONSFind an unknown length on one of two similar figuresFOUNDATIONSCompute a shorter side from the hypotenuse and the other legFOUNDATIONS
Inverse circular derivatives
Second derivatives and graph reasoning
Implicit differentiation and related rates