Skip to content
Home
Path
Worksheets
Skill map
Leaderboard
Sign in
Save your progress
Search topics
Ctrl K
Sign in
Save your progress
Search
Skill map
Mathematical Methods
Differentiation
Chapter 5
0 of 56 strong
Weak only
Not started
RESTS ON
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 factor
CH 1
Identify the asymptotes, excluded domain and range values, branch orientation, intercepts and end behaviour of a translated and dilated reciprocal function
CH 2
Evaluate and analyse a general piecewise-defined function with non-linear branches, including branch selection and junction behaviour
CH 12
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 endpoints
CH 2
Signed numbers
CH 3
Multiplication and division
CH 9
Remove the greatest common numerical and algebraic factor from every term before considering any further factorisation
CH 6
Factor a simple quadratic with integer coefficients by finding factor data whose product and sum reproduce the constant and linear coefficients
CH 3
Recognise and factor a difference of two squares, a perfect-square trinomial, and a sum or difference of two cubes, with the correct sign pattern
FOUNDATIONS
Decimal place value
FOUNDATIONS
Solve a one-step linear equation
CH 7
Simplify products, quotients and powers with a common positive base by adding, subtracting or multiplying exponents as appropriate, without applying an index law across addition or subtraction
CH 7
Extract maximal perfect-power factors from a radicand and place the result in canonical coefficient-times-squarefree-radical form
CH 2
Connect a root of P(x)=0, its linear factor and the corresponding horizontal-axis intercept of y=P(x)
FOUNDATIONS
Compute a slope from a graph as rise over run between two points
FOUNDATIONS
Identify the intercept and slope in y = a + bx, including its edge shapes
CH 2
Classify a graphed line's slope as positive, negative, zero or undefined
FOUNDATIONS
Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powers
CH 1
Multiply polynomials using complete distribution and exponent laws, then collect equal powers
CH 1
Solve equations with unknowns on both sides by collecting them onto one side
CH 1
Check a claimed solution or point by substituting into the original equation
CH 1
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 approximation
CH 4
Turn an equation's exact solution into the answer the situation demands
CH 4
Identify growth or decay, horizontal asymptote, intercepts, domain, range, one-to-one status and end behaviour of a basic or transformed exponential graph
CH 2
Percentage change as a multiplier
FOUNDATIONS
Apply reflection, parity and periodicity relations to rewrite a circular-function value in terms of a base angle and the correct sign
CH 1
Reduce an angle to its first-quadrant reference angle measured from the horizontal axis
CH 1
Determine which of sine, cosine and tangent are positive, negative, zero or undefined in each quadrant or on a boundary
CH 1
Reduce an extended positive or negative rotation and identify its quadrant or axis boundary
CH 1
Name each side of a right-angled triangle relative to a marked angle
FOUNDATIONS
Compute the perimeter of a described polygon as the sum of its side lengths
FOUNDATIONS
Differentiate a constant multiple, sum or difference term by term while preserving each exact rational coefficient and sign
FOUNDATIONS
Write a recurrence relation as a starting value and a rule
FOUNDATIONS
Establish that at least one real root lies in an interval by exact endpoint evaluation with opposite signs, without claiming uniqueness or claiming the converse when the signs agree
FOUNDATIONS
Limits, first principles and the power rule
Gradient graphs and smooth joins
Tangents, normals and derivative conditions
Exponential derivatives and rate models
Trigonometric differentiation and rates
Newton’s method and numerical accuracy
Evaluate a finite limit by exact substitution when the displayed rule is defined at the approach value, or by an exact algebraic simplification that removes the indeterminate form before substitution; for a rational rule at an unbounded input, compare degrees and leading coefficients to obtain any finite limiting value
Determine the exact left and right limiting values of a piecewise rule or supplied graph and use their equality or inequality to decide whether the two-sided finite limit exists
Test the three continuity conditions at a stated input: the function value exists, the two-sided limit exists, and the two values agree
Form and simplify the secant-gradient difference quotient (f(x+h)−f(x))/h for a polynomial or simple power function, expanding f(x+h) exactly and cancelling the common factor of h before any limit is taken
Take the limit of a simplified difference quotient as h→0 to state the derivative rule, and use it to find the exact gradient at a named point, substituting the named x-value only after the limit has been taken
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 required
Substitute an exact value into a derivative and compute the exact rational gradient, distinguishing f'(a) from f(a) and from the unsimplified derivative rule
Solve f'(x)=c or f'(x)=0 over a stated domain for a polynomial or power derivative, using inverse operations and rejecting solutions outside the domain
Apply the distributive law repeatedly, preserve signs and collect like powers to write a polynomial in descending-power form
Factor numerators and denominators, cancel common factors rather than terms, retain the original excluded values, and present the remaining numerator and denominator in a consistent form
Multiply and divide fractions
Simplify a fraction
Evaluate P(a) by substituting an integer, exact rational or simple exact irrational value into every occurrence of the variable
Substitute known values into a formula and evaluate it correctly
Solve two-step linear equations by inverse operations in reverse order
Convert exactly between negative or rational powers and reciprocal radical forms, and evaluate constructions whose required roots are exact
Decide whether a function is differentiable at a point by checking continuity and equality of the exact one-sided derivatives, distinguishing a smooth join from a corner, cusp, vertical tangent or discontinuity
Determine exact parameters that make polynomial branches continuous and differentiable at one or more joins by solving value-matching and derivative-matching equations together
Translate 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 derivative
Determine horizontal- and vertical-axis intercepts, a stated or structurally visible stationary point of inflection, and qualitative concavity change from a cubic rule or graph
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)=c
Compute the point of contact and evaluate the derivative at that x-value for the gradient, then write the tangent in y=mx+c or ax+by=c form from the point and gradient
Form the perpendicular line to a tangent through the same point of contact by taking the negative reciprocal of the tangent gradient, excluding the zero-gradient case where no finite negative reciprocal exists
Use a stated parallel or perpendicular line, or a stated tangent gradient, to locate the point of tangency or an unknown parameter, and complete the required line
Recover a polynomial rule from its derivative and enough point or tangent conditions by reversing the power rule term-by-term and determining the constant of anti-differentiation from the supplied data
Compute a slope from two coordinate pairs with the slope formula
Write a line's equation from its y-intercept and slope
Use equal gradients for parallel oblique lines and product negative one for perpendicular oblique lines, with explicit horizontal–vertical edge cases
Recognise that an identity holds for every input and determine unknown parameters by expanding and equating coefficients of like powers
Solve a pair of simultaneous linear equations for the shared (x, y)
Rounding
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
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
For 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
Solve for a time, doubling time or half-life from an exponential model, respecting the model domain, the stated time unit and exact-versus-approximate answer requirements
Evaluate a stated exponential model at an admissible time and interpret the predicted amount or change without rounding an intermediate power
Interpret the initial value, per-period multiplier, growth or decay rate, horizontal asymptote or long-run value, and the time unit in a model of the form L+A b^t
Differentiate sine, cosine and tangent of kx and g(x), converting degree arguments to radians first, simplifying with the Pythagorean and tangent identities where possible, and using the reciprocal-square form for tangent
Evaluate a trigonometric derivative at standard angles, solve derivative equations such as f'(x)=0 or f'(x)=c over a finite interval, and retain exactly the solutions in the stated interval
Build an area, length or height expression in terms of an angle from a described geometric figure, using exact trigonometric side relations and standard area formulas, and verify a proposed expression coefficient-by-coefficient
Read amplitude, equilibrium level, period and phase from a sinusoidal model; compute values and exact derivative values at standard-angle times; solve rate equations and identify intervals of increase or decrease
Differentiate a trigonometric model, solve the stationary condition exactly at standard angles, classify the stationary point as a maximum or minimum by the model or interval, and evaluate the optimum in context
Convert an angle between degrees and radians, keeping an exact rational multiple of π where available and distinguishing a degree label from an unlabelled radian value
Use sin²(x)+cos²(x)=1 and tan(x)=sin(x)/cos(x) to replace, simplify or verify an expression on its valid domain
Obtain tan θ = y/x from a unit-circle point and recognise that tangent is undefined when x = 0
Find all angles in a stated finite domain whose sine, cosine or tangent has a given exact value, using reference angles and periodicity
Obtain exact sine, cosine and tangent values for the standard first-quadrant angles, their symmetric points and axis boundaries
Use the terminal point (x, y) on the unit circle to obtain cos θ = x and sin θ = y, or recover the coordinate pair from known sine and cosine values
State and apply the maximal domains and ranges of the sine, cosine and tangent functions, including tangent's excluded odd half-turn boundaries
Translate a practical description into a labelled right-triangle model
Select and apply the area formula matching the named shape and measurements
Apply the Newton recurrence to a cubic polynomial using the current iterate in both the function and derivative, producing each subsequent iterate from exact rational arithmetic
Use certified bracketing and stable rounded iterates to report the intended cubic root to a stated precision, while recognising starting values or derivative-zero steps for which the iteration is unsuitable
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 them
Iterate a recurrence relation to generate terms
Repeatedly bisect a sign-changing interval, select the subinterval whose endpoint values have opposite signs, and use a rigorous interval-width rule to state a root approximation