Differentiation — Edexcel A-Level Mathematics
Test yourself on Differentiation with PEARSON EDEXCEL A-Level practice questions.
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Differentiation explained
The derivative f′(x) gives the gradient of the tangent to y = f(x) at the general point (x, y).
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It is defined as the limit of [f(x + h) − f(x)] ÷ h as h tends to zero. This limit is the gradient of the chord as the two points merge. The derivative also represents a rate of change, such as velocity from displacement. Sketching the gradient function means plotting where f′(x) is positive, negative or zero, matching stationary points to axis crossings. Differentiating from first principles works for small positive integer powers of x and for sin x and cos x. The second derivative f″(x) is the rate of change of the gradient; where f″(x) > 0 the curve is convex, where f″(x) < 0 it is concave, and a change of sign indicates a point of inflection.
7.2 Differentiate xⁿ, for rational values of n, and related constant multiples, sums and differences. Differentiate e^(kx) and a^(kx), sin kx, cos kx, tan kx and related sums, differences and constant multiples. Understand and use the derivative of ln x as 1/x.
This section builds the core library of derivatives you apply throughout calculus. For a power xⁿ with rational n, the derivative is n x^(n−1); this works for positive, negative and fractional indices, so d/dx(x^(1/2)) = (1/2)x^(−1/2) and d/dx(x^(−2)) = −2x^(−3). Constant multiples, sums and differences differentiate term by term: if y = 5x³ − 2x^(1/2) + 7, then dy/dx = 15x² − x^(−1/2). For exponentials, d/dx(e^(kx)) = k e^(kx), and d/dx(a^(kx)) = k a^(kx) ln a, so d/dx(2^(3x)) = 3·2^(3x) ln 2. For trigonometric functions with argument kx, d/dx(sin kx) = k cos kx, d/dx(cos kx) = −k sin kx and d/dx(tan kx) = k sec² kx. The natural logarithm has d/dx(ln x) = 1/x for x > 0. Combine these rules with constant multiples and sums, for example d/dx(4 sin 3x − 2e^(5x)) = 12 cos 3x − 10e^(5x).
7.3 Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection. Identify where functions are increasing or decreasing.
Differentiation gives the gradient function, which you use to describe curve behaviour. To find the gradient at a point, substitute the x-coordinate into dy/dx. The tangent at that point has gradient m = dy/dx, and its equation follows from y − y₁ = m(x − x₁). The normal is perpendicular, so its gradient is −1/m when m ≠ 0. Stationary points occur where dy/dx = 0; classify them by the sign of the second derivative: d²y/dx² > 0 gives a minimum, d²y/dx² < 0 gives a maximum, and d²y/dx² = 0 requires further checking because it may be a point of inflection. A point of inflection is where the curve changes concavity, often with d²y/dx² = 0 and a change in sign of d²y/dx². A function is increasing where dy/dx > 0 and decreasing where dy/dx < 0, so solve the resulting inequality to identify intervals.
7.4 Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.
Composite and combined functions need structured rules. The product rule states that if y = uv, then dy/dx = u dv/dx + v du/dx. The quotient rule states that if y = u/v, then dy/dx = (v du/dx − u dv/dx)/v². The chain rule handles composition: if y = f(g(x)), then dy/dx = f′(g(x))·g′(x), often written as dy/dx = dy/du × du/dx. These rules extend to connected rates of change, where dy/dt = dy/dx × dx/dt links how quantities change together, and to inverse functions, where dx/dy = 1/(dy/dx) provided dy/dx ≠ 0. For example, differentiating y = x² sin x uses the product rule to give 2x sin x + x² cos x, while y = (3x + 1)⁵ uses the chain rule to give 15(3x + 1)⁴.
7.5 Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.
Some curves are easier to describe implicitly or parametrically. For implicit relations, differentiate each term with respect to x, remembering that y is a function of x, so d/dx(yⁿ) = n y^(n−1) dy/dx by the chain rule. For example, differentiating x² + y² = 25 gives 2x + 2y dy/dx = 0, so dy/dx = −x/y. Products such as xy require the product rule, giving x dy/dx + y. For parametric curves, x and y are given in terms of a parameter t; the first derivative is dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. For example, if x = t² and y = t³, then dy/dx = 3t²/(2t) = 3t/2. Only the first derivative is required here, so you do not need to find d²y/dx² for these implicit or parametric cases.
7.6 Construct simple differential equations in pure mathematics and in context (contexts may include kinematics, population growth and modelling the relationship between price and demand).
Constructing a differential equation means translating a written rate relationship into an equation containing a derivative. Identify the two quantities that change, choose symbols, then write the stated rate as a derivative: for example, if a population P grows at a rate proportional to its current size, write dP/dt = kP. In kinematics, acceleration is the rate of change of velocity, so a = dv/dt, and velocity is the rate of change of displacement, so v = ds/dt; a constant deceleration of 2 m s⁻² gives dv/dt = −2. For price and demand, if demand x falls at a rate proportional to its current value, dx/dt = −kx. Define every symbol, state whether each constant is positive, and keep units consistent.
Your focus
- Derive derivatives from first principles for small positive integer powers of x and for sin x and cos x.
- Interpret the derivative as a gradient and as a rate of change.
- Use the second derivative to classify convex and concave sections and identify points of inflection.
Show all 18 objectives
- Differentiate expressions of the form axⁿ with rational n, including sums, differences and constant multiples.
- Differentiate e^(kx), a^(kx), sin kx, cos kx and tan kx accurately.
- Apply the derivative of ln x as 1/x within combined expressions.
- Calculate gradients and form equations of tangents and normals at given points.
- Locate and classify stationary points using first and second derivatives.
- Determine intervals where a function is increasing or decreasing and recognise points of inflection.
- Differentiate products and quotients of functions using the product and quotient rules.
- Differentiate composite functions using the chain rule.
- Solve connected rates of change problems and differentiate inverse functions using reciprocal relationships.
- Differentiate implicit relations to find dy/dx in terms of x and y.
- Differentiate parametric equations to find dy/dx in terms of the parameter.
- Evaluate gradients at given points or parameter values for implicit and parametric curves.
- Translate a written rate statement into a differential equation using correct derivative notation.
- Select and define appropriate variables and constants for a given context.
- Justify the sign and meaning of each term in a constructed differential equation.
Differentiation exam tips
Marking Points
- States the derivative as the limit of [f(x + h) − f(x)] ÷ h as h tends to zero.
- Interprets f′(x) as the gradient of the tangent at a general point and as an instantaneous rate of change.
- Sketches the gradient function by identifying where f′(x) is positive, negative and zero relative to the original curve.
- Differentiates small positive integer powers of x from first principles by expanding and taking the limit.
- Derives the derivatives of sin x and cos x from first principles using small-angle approximations or limit results.
- Uses the second derivative as the rate of change of gradient and links its sign to convex and concave sections and points of inflection.
- Applies the power rule n x^(n−1) correctly to rational indices, including negative and fractional powers such as x^(−3) and x^(2/3).
- Differentiates constant multiples, sums and differences term by term, preserving coefficients and signs.
- States and uses d/dx(e^(kx)) = k e^(kx) and d/dx(a^(kx)) = k a^(kx) ln a with the correct constant k and base factor.
- Differentiates sin kx, cos kx and tan kx as k cos kx, −k sin kx and k sec² kx respectively.
- Uses d/dx(ln x) = 1/x, including within sums and constant multiples such as 7 ln x.
- Substitutes a given x-value into dy/dx to obtain the gradient at a point and uses it in the tangent equation.
- Finds the normal gradient as the negative reciprocal of the tangent gradient and forms the correct line equation.
- Solves dy/dx = 0 to locate stationary points and gives both coordinates.
- Uses the second derivative, or a sign test on dy/dx, to classify maxima, minima and points of inflection.
- Determines intervals where dy/dx > 0 or dy/dx < 0 to state where the function is increasing or decreasing.
- Identifies the correct rule for a product, quotient or composite function and states the rule before substituting.
- Applies the product rule as u dv/dx + v du/dx with correct differentiation of each factor.
- Applies the quotient rule with the numerator v du/dx − u dv/dx and denominator v², keeping the subtraction order correct.
- Uses the chain rule to differentiate nested functions, including powers of linear expressions and trigonometric composites.
- Sets up and solves connected rates of change using dy/dt = dy/dx × dx/dt, and uses dx/dy = 1/(dy/dx) for inverse functions.
- Differentiates implicit terms in y with respect to x, attaching dy/dx through the chain rule.
- Applies the product rule to mixed terms such as xy when differentiating implicitly.
- Rearranges the differentiated equation to make dy/dx the subject, factorising where necessary.
- Finds dx/dt and dy/dt for parametric equations and forms dy/dx as their quotient.
- Substitutes a given point or parameter value to obtain a numerical gradient where required.
- Correctly identifies the dependent variable and the independent variable before differentiating, for example population P with respect to time t.
- Writes the given rate as a derivative with correct notation, such as dP/dt, dv/dt or dx/dt, rather than as a ratio of small changes.
- Translates proportional relationships using a constant of proportionality, for example dP/dt = kP, and states that k is a positive constant.
- Uses correct signs for increase or decrease, for example dv/dt = −2 for deceleration and dx/dt = −kx for falling demand.
- Defines all symbols and, where required, gives the units of the constant of proportionality so the equation is dimensionally consistent.
- Handles related rates in context by linking two derivatives through a chain rule relationship when the statement requires it.
Examiner Tips
- 💡Write the limit definition in full before simplifying, so the first-principles method is clear.
- 💡When sketching a gradient function, mark axis crossings and turning points of the original curve first.
- 💡Justify points of inflection by showing a sign change in f″(x), not just f″(x) = 0.
- 💡Rewrite roots and reciprocals as fractional or negative indices before differentiating, for example √x as x^(1/2).
- 💡Check each term separately, then combine, so a sign error in one term does not spread through the whole expression.
- 💡When a base is not e, such as 5^(2x), include ln 5 as a factor rather than leaving the derivative as a multiple of the original function alone.
- 💡Write the gradient function clearly before substituting values, so arithmetic errors are easier to spot.
- 💡For increasing or decreasing intervals, solve the inequality and express the answer as a range of x-values rather than a single point.
- 💡When classifying a stationary point, show the second derivative value or a sign table to justify the conclusion.
- 💡Label u and v, or the inner and outer functions, before differentiating to reduce substitution errors.
- 💡For connected rates, write the chain relationship explicitly before inserting numerical values.
- 💡Check whether a rate is positive or negative in context, especially when a quantity is decreasing.
- 💡Collect all dy/dx terms on one side before factorising to isolate the derivative.
- 💡For parametric questions, write dx/dt and dy/dt separately before forming the quotient.
- 💡Substitute the given coordinates or parameter value only after obtaining the general expression for dy/dx.
- 💡Underline the rate words in the question, such as rate of change, proportional to or per unit time, before writing anything.
- 💡Define each symbol in a short sentence so the examiner can see your model clearly.
- 💡Check the sign and units of your constant against the context before moving on.
Common Mistakes
- Treating the derivative as the gradient of a chord rather than the tangent; correct this by taking the limit as h tends to zero.
- Assuming any point where f″(x) = 0 is a point of inflection; correct this by checking that the concavity changes sign.
- Confusing convex and concave: convex means f″(x) > 0 and concave means f″(x) < 0; correct this by testing a value on each side.
- Forgetting to multiply by the inner constant k when differentiating e^(kx), sin kx or cos kx; correct by writing the factor k explicitly before the function.
- Subtracting one from the coefficient instead of the index when differentiating a power; correct by reducing only the exponent, as in x⁵ becoming 5x⁴.
- Treating ln x as differentiating to ln x or to x^(−1) written incorrectly; correct by writing the derivative as 1/x, valid for x > 0.
- Using the tangent gradient directly for the normal; correct by taking the negative reciprocal, so a tangent gradient of 3 gives a normal gradient of −1/3.
- Assuming every point with d²y/dx² = 0 is a point of inflection; correct by checking that concavity actually changes sign.
- Giving only the x-coordinate of a stationary point; correct by substituting back into the original equation to find the y-coordinate.
- Reversing the subtraction in the quotient rule; correct by writing v du/dx first and subtracting u dv/dx.
- Differentiating a product by multiplying the separate derivatives; correct by using u dv/dx + v du/dx.
- Forgetting the inner derivative in the chain rule; correct by multiplying by the derivative of the inside function.
- Differentiating y terms as if y were a constant; correct by including dy/dx for every differentiated y term.
- Forgetting the product rule on terms such as xy; correct by writing x dy/dx + y.
- Inverting the parametric quotient; correct by dividing dy/dt by dx/dt, not the other way round.
- Writing the rate as a plain fraction of changes instead of a derivative; correct this by using dP/dt or dv/dt notation.
- Omitting the constant of proportionality in a proportional relationship; correct this by including k and stating its sign.
- Using the wrong sign for a decreasing quantity; correct this by checking whether the context says increase or decrease and inserting a negative sign where needed.