Logarithmic calculus

Chapter 90 of 60 strong

RESTS ONInterpret the derivative as the limit of a simplified difference quotient as the increment approaches zero, cancelling the common increment only after it has been factored and never treating the original quotient as defined at zeroFOUNDATIONSInterpret (f∘g)(x) as f(g(x)), identify the inner and outer function and the order of application, and distinguish composition from multiplication and from the reversed compositionCH 3 Simplify 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 themCH 5 Rewrite reciprocal, root and rational terms as a constant times x^n, then differentiate term-by-term with the power rule n a x^(n−1), using linearity for sums and differences and returning the result with positive or fractional indices as requiredCH 5 Differentiate a monomial with non-negative integer exponent by multiplying by the exponent and reducing that exponent by one, with a constant term differentiating to zeroCH 6 Extract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical formCH 2 Obtain the domain and range of a relation represented by ordered pairs, a table, a rule or described graph data, preserving set membership and open or closed endpointsCH 2 Factor a simple quadratic with integer coefficients by finding factor data whose product and sum reproduce the constant and linear coefficientsCH 3Recognise and factor a difference of two squares, a perfect-square trinomial, and a sum or difference of two cubes, with the correct sign patternFOUNDATIONSHighest common factorFOUNDATIONSSubstitute known values into a formula and evaluate it correctlyCH 4 Find the equation of the line through two pointsCH 1 Relate an original polynomial graph to its derivative graph through tangent-gradient sign, zeros of the derivative, degree reduction and relative steepness, in either direction where the supplied information is sufficientCH 6 Calculate the average rate of change of a function over a non-degenerate interval as the exact change in output divided by the exact change in input, from a rule, table, two points or a factual context, keeping the subtraction order matched between the output change and the input changeCH 6 Apply the unitary method to a rateCH 7 Identify endpoints or central points, domain, range, orientation and correspondence for square-root and cube-root functions and their simple transformationsCH 2 Identify the intercept and slope in y = a + bx, including its edge shapesCH 2 Compute a slope from two coordinate pairs with the slope formulaCH 2Classify a graphed line's slope as positive, negative, zero or undefinedFOUNDATIONSSolve a one-step linear equationCH 7 Completely factorise a low-degree polynomial by taking out common factors, using familiar quadratic patterns, or finding one linear factor and dividing to obtain a lower-degree factorCH 1 Solve equations with unknowns on both sides by collecting them onto one sideCH 1 Determine both end directions of a polynomial from degree parity and the sign of its leading coefficientCH 2Use the parity and size of a root's multiplicity to determine whether the graph crosses or touches the horizontal axis, distinguish an ordinary crossing from a stationary crossing, and decide whether the sign changesFOUNDATIONSAnti-differentiate a x^n by increasing the exponent by one and dividing the exact coefficient by that new exponent, with the stated exclusion n = -1CH 7 Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powersCH 1 Identify integrand, variable of integration, lower endpoint and upper endpoint, and distinguish a definite integral's numeric value from an indefinite integral's family of functionsCH 8 Classify a relation as one-to-one, many-to-one, one-to-many or many-to-many from repeated inputs and outputs or from horizontal- and vertical-line evidenceCH 2 Construct the inverse of a relation by interchanging every ordered pair, hence interchanging domain and range, and map graph points through reflection in y=xCH 3 Solve an exponential equation that cannot be reduced to a common base by taking logarithms consistently, isolate the unknown and distinguish the exact logarithmic form from a guarded decimal approximationCH 4 Turn an equation's exact solution into the answer the situation demandsCH 4 Identify growth or decay, horizontal asymptote, intercepts, domain, range, one-to-one status and end behaviour of a basic or transformed exponential graphCH 2Percentage change as a multiplierFOUNDATIONSCompute the perimeter of a described polygon as the sum of its side lengthsFOUNDATIONS

Logarithmic differentiation

Applications of logarithmic derivatives

Reciprocal and linear-reciprocal antiderivatives

Recognising logarithmic derivatives

Areas, inverse functions and logarithmic models

Recall and apply the natural-logarithm derivative in its base forms — d/dx log_e(kx) = 1/x and d/dx log_e(g(x)) = g'(x)/g(x) — identifying the inner function g, using log laws first where that simplifies, and stating the domain on which the rule applies

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Practise this

RESTS ON

Identify 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 formNot startedConnect f'(x), dy/dx, a derivative with respect to a named independent variable, and evaluation notation at a stated input, while treating each as notation for a function or value rather than ordinary fraction arithmeticNot startedFactor numerators and denominators, cancel common factors rather than terms, retain the original excluded values, and present the remaining numerator and denominator in a consistent formNot startedForm a function's maximal real domain by excluding zeros of denominators, requiring radicands to be non-negative, requiring denominator radicands to be positive, and intersecting all component restrictionsNot started

UNLOCKS

Differentiate products, quotients and rewritten forms in which at least one component is logarithmic, selecting the product rule, quotient rule, chain rule or a log-law simplification first, and reporting the derivative in the requested factorised or canonical formDetermine and preserve the maximal domain on which a logarithmic expression, its simplification and its derivative are defined, recognising domain changes caused by log-law rewriting, and state the derivative together with its restrictionsApply logarithmic derivatives to a gradient at a point, a tangent or normal equation, or a stationary point of a function involving logarithms, solving exact derivative equations such as log_e(ax+b) = c by the inverse exponential relation and classifying the stationary pointAnti-differentiate k/x and k/(ax+b) to the logarithmic form on the valid branch (or with absolute value where an interval lies on the negative side of the singularity), and evaluate definite integrals exactly in terms of log_eAnti-differentiate by recognition where the supplied derivative contains logarithmic terms, recognise the ∫ g'/g dx = log_e|g| pattern, then read off the requested anti-derivative or evaluate a related definite integralInterpret and apply a logarithmic model in context, evaluating the model, finding an instantaneous rate by differentiation, solving for an input by the inverse exponential relation, and reporting correct units and sign over the model's stated domain