Applications of differentiation

Chapter 60 of 57 strong

RESTS ONRewrite 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 Interpret f, f(x), image, preimage, rule, domain and co-domain; evaluate a function at a stated exact input and solve a simple equation of the form f(x)=cCH 2Interpret 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 zeroFOUNDATIONSObtain the derivative of a low-degree polynomial from the limit definition and evaluate the resulting derivative at an exact input when askedFOUNDATIONSSimplify a fractionCH 1Highest common factorFOUNDATIONSDifferentiate a constant multiple, sum or difference term by term while preserving each exact rational coefficient and signFOUNDATIONSRearrange a polynomial equation to zero, factorise it, and use the null factor law to obtain all real solutionsCH 1 Find the equation of the line through two pointsCH 1 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 Solve two-step linear equations by inverse operations in reverse orderCH 1Recognise and factor a difference of two squares, a perfect-square trinomial, and a sum or difference of two cubes, with the correct sign patternFOUNDATIONSMultiply and divide fractionsCH 1 Signed numbersCH 3 Multiplication and divisionCH 9 Identify the intercept and slope in y = a + bx, including its edge shapesCH 2 Determine the sign of a factorised polynomial on intervals separated by its ordered real roots, changing sign exactly at odd-multiplicity rootsCH 2 Solve a pair of simultaneous linear equations for the shared (x, y)CH 1Classify a graphed line's slope as positive, negative, zero or undefinedFOUNDATIONSCompute a slope from two coordinate pairs with the slope formulaCH 2Form or recognise the union, intersection, difference and complement of finite or interval-defined sets relative to an explicitly stated universeFOUNDATIONSDetermine 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 changesFOUNDATIONSConnect a root of P(x)=0, its linear factor and the corresponding horizontal-axis intercept of y=P(x)FOUNDATIONSDecide whether an algebraic expression is a polynomial in the named variable by checking that every variable exponent is a non-negative integer; coefficients may be any real numbersFOUNDATIONSConnect zero remainder, P(a)=0, the linear factor corresponding to a, and the zero a, in both directionsCH 1 Divide one polynomial by a lower-degree polynomial, maintaining descending-power columns and zero placeholders, to obtain an exact quotient and a remainder of lower degreeCH 1 Multiply polynomials using complete distribution and exponent laws, then collect equal powersCH 1 Substitute known values into a formula and evaluate it correctlyCH 4 Solve a one-step linear equationCH 7 Solve equations with unknowns on both sides by collecting them onto one sideCH 1 Apply the unitary method to a rateCH 7Compute the scale factor k of a similar pairFOUNDATIONSIdentify the hypotenuse and compute it from the two shorter sidesCH 1Compute a circle's circumference and area from a stated radius or diameterFOUNDATIONSCompute a prism's volume as cross-sectional area × lengthFOUNDATIONSCompute the perimeter of a described polygon as the sum of its side lengthsFOUNDATIONS

Basic derivatives

Differentiation rules

Tangents, normals and curve sketching

Optimisation

Rates of change

Kinematics

Differentiate e^x, e^(kx) and e^(g(x)) with the factor of the inner derivative, and differentiate sums, expanded squares and quotient forms after rewriting them as sums of e^(kx) terms

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RESTS ON

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 requiredNot 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 started

UNLOCKS

Evaluate an exponential derivative at an exact point, solve f'(x)=c by isolating the exponential and applying the natural-logarithm inverse, and state whether the solution is exact or approximate CH 5For a model of the form A=A0 e^(kt), identify the initial amount, compute amount and rate of change at a stated time, recover k from two exact model values, solve a threshold-time equation, and interpret the sign and units of the derivative CH 5Identify 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 formIdentify 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 formAnti-differentiate a·e^(kx) as (a/k)·e^(kx) for non-zero exact k, and a linear combination of such terms term-by-term, verifying by differentiation CH 7Given d/dx[F(x)] = f(x), conclude ∫ f(x) dx = F(x) + c, and use that derivative to anti-differentiate a stated exact multiple of f(x), matching the chain-rule factor rather than integrating the visible factors separately CH 7