G: Differentiation — AQA A-Level Mathematics
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G: Differentiation explained
The derivative f'(x) gives the gradient of the tangent to y = f(x) at a 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 0. This limit is the gradient of the tangent, and it also represents an instantaneous rate of change, such as velocity or marginal cost. You can sketch the gradient function by noting where f is increasing (f' > 0), decreasing (f' < 0), and stationary (f' = 0). The second derivative f''(x) is the derivative of f'(x); it measures how the gradient changes. Where f'' > 0 the curve is convex (concave up), where f'' < 0 it is concave (concave down), and a point of inflection occurs where the concavity changes, often when f'' = 0 and changes sign. You must differentiate from first principles for small positive integer powers of x and for sin x and cos x.
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 topic covers the standard derivatives required at A-level. For powers, use d/dx(x^n)=n x^{n-1} for any rational n, including negative and fractional indices. Apply the constant multiple, sum and difference rules term by term. For exponentials, d/dx(e^{kx})=k e^{kx} and d/dx(a^{kx})=k a^{kx} ln a. For trigonometric functions, d/dx(sin kx)=k cos kx, d/dx(cos kx)=-k sin kx, and d/dx(tan kx)=k sec^2 kx. Combine these with sums, differences and constant multiples. Also know that d/dx(ln x)=1/x. For example, differentiate 3x^{1/2} - 2e^{3x} + 5 sin 2x: the derivative is (3/2)x^{-1/2} - 6e^{3x} + 10 cos 2x. This skill underpins later calculus work.
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. To find the gradient at a point, substitute the x-coordinate into dy/dx. The tangent at that point has equation y - y1 = m(x - x1), where m is the gradient. The normal is perpendicular, so its gradient is -1/m (provided m ≠ 0). Stationary points occur where dy/dx = 0; solve this equation. Classify them using the second derivative: if d²y/dx² > 0, it's a minimum; if < 0, a maximum; if = 0, further investigation is needed. Points of inflection occur where the concavity changes, often when d²y/dx² = 0 and changes sign. A function is increasing where dy/dx > 0 and decreasing where dy/dx < 0. For example, for y = x³ - 3x, dy/dx = 3x² - 3; stationary points at x = ±1; d²y/dx² = 6x, so x=1 is a minimum and x=-1 a maximum.
Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.
This topic extends differentiation beyond simple polynomials. The product rule differentiates y = uv as u dv/dx + v du/dx. The quotient rule differentiates y = u/v as (v du/dx − u dv/dx)/v², where du/dx is the derivative of the numerator u and dv/dx is the derivative of the denominator v. The chain rule differentiates composite functions: dy/dx = dy/du × du/dx. For connected rates, use dy/dt = dy/dx × dx/dt, often with geometry formulae. For inverse functions, dy/dx = 1/(dx/dy), provided dx/dy ≠ 0. For example, differentiate y = x² sin x using the product rule, y = ln x / x using the quotient rule, and y = (3x + 1)^5 using the chain rule. Then solve a related-rates problem such as an expanding circle, and differentiate an inverse by finding dx/dy first.
Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.
This topic covers differentiation when y is not given explicitly as a function of x. For implicit relations, differentiate each term with respect to x, applying the chain rule to terms containing y: d/dx[f(y)] = f'(y) dy/dx. Then rearrange to make dy/dx the subject. For parametric equations x = x(t), y = y(t), use dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0. For example, for x² + y² = 25, differentiate to get 2x + 2y dy/dx = 0, so dy/dx = −x/y. For x = t², y = 2t, dy/dx = (2)/(2t) = 1/t. Only first derivatives are required.
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 or physical description into an equation involving a derivative. Identify the changing quantities and the independent variable, then express the given rate of change 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 dv/dt or d²x/dt². For price-demand, if demand falls at a rate proportional to the excess of price over a threshold, write dD/dt = -k(P - P0). Always define symbols and include the constant of proportionality. The equation should capture the relationship, not solve it. In pure mathematics, you may be given a family of curves and asked to form the differential equation by eliminating parameters. The skill is assessed by checking that the derivative is correctly identified and the relationship is accurately represented.
Your focus
- Students can explain the derivative as a limit and use first principles to differentiate simple functions.
- Students can sketch the gradient function for a given curve and interpret the derivative as a rate of change.
- Students can use the second derivative to determine convexity, concavity, and points of inflection.
Show all 21 objectives
- Differentiate x^n for rational n and related constant multiples, sums and differences.
- Differentiate e^{kx}, 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.
- Apply differentiation to find gradients, tangents and normals.
- Find and classify maxima, minima and stationary points, and identify points of inflection.
- Identify where functions are increasing or decreasing.
- Apply the product rule to differentiate products of functions.
- Apply the quotient rule to differentiate quotients of functions.
- Apply the chain rule to differentiate composite functions.
- Solve problems involving connected rates of change using the chain rule.
- Differentiate inverse functions using the relationship dy/dx = 1/(dx/dy).
- Differentiate implicit relations to find dy/dx.
- Differentiate parametric equations to find dy/dx.
- Evaluate the gradient at a given point for implicit or parametric curves.
- Rearrange equations to make dy/dx the subject.
- Translate a written description of a changing quantity into a differential equation.
- Identify the appropriate derivative and independent variable in a given context.
- Form a differential equation from a family of curves by eliminating parameters.
G: Differentiation exam tips
Marking Points
- Interpret the derivative as the gradient of the tangent and as an instantaneous rate of change in context.
- Use the limit definition to differentiate from first principles for functions such as x^n (small positive integer n), sin x, and cos x.
- Sketch the graph of the gradient function for a given curve, identifying where the derivative is positive, negative, or zero.
- Calculate second derivatives and interpret them as the rate of change of the gradient.
- Use the sign of the second derivative to determine convexity/concavity and identify points of inflection.
- Apply the power rule d/dx(x^n)=n x^{n-1} correctly for rational n, including negative and fractional indices.
- Differentiate constant multiples, sums and differences term by term, preserving coefficients.
- Differentiate e^{kx} as k e^{kx} and a^{kx} as k a^{kx} ln a.
- Differentiate sin kx, cos kx and tan kx as k cos kx, -k sin kx and k sec^2 kx respectively.
- State and use the derivative of ln x as 1/x, recognising that ln x is not differentiated by the power rule.
- Find the gradient of a curve at a given point by evaluating dy/dx at that point.
- Determine the equation of the tangent or normal to a curve at a given point.
- Locate stationary points by solving dy/dx = 0 and classify them using the second derivative or sign of dy/dx.
- Identify points of inflection where the concavity changes, often using d²y/dx² = 0 and sign change.
- Determine intervals where a function is increasing (dy/dx > 0) or decreasing (dy/dx < 0).
- Correctly identify the structure of the function: product, quotient, composite, connected rates or inverse, before choosing a rule.
- Apply the product rule accurately: for y = uv, dy/dx = u dv/dx + v du/dx, with correct derivatives of u and v.
- Apply the quotient rule accurately: for y = u/v, dy/dx = (v du/dx − u dv/dx)/v², maintaining the correct order in the numerator: the derivative of the numerator u appears first with a positive sign, and the derivative of the denominator v appears second with a negative sign.
- Apply the chain rule accurately: for y = f(g(x)), dy/dx = f'(g(x)) × g'(x), or in Leibniz notation dy/dx = dy/du × du/dx.
- For connected rates of change, set up a chain such as dV/dt = dV/dr × dr/dt and substitute known rates and dimensions correctly.
- For inverse functions, use dy/dx = 1/(dx/dy), ensuring the derivative is evaluated at the corresponding point and that dx/dy is not zero.
- Differentiate implicit relations term by term with respect to x, remembering to multiply by dy/dx when differentiating a function of y.
- Rearrange the resulting equation correctly to isolate dy/dx, including collecting like terms and factorising if necessary.
- For parametric equations, differentiate x and y with respect to the parameter t to find dx/dt and dy/dt.
- Form the first derivative as dy/dx = (dy/dt)/(dx/dt), ensuring the correct order of division.
- Substitute a given parameter value or point to evaluate the gradient when required.
- Recognise when a relation is defined implicitly or parametrically and choose the appropriate method.
- Correctly identify the dependent and independent variables from the context.
- Express the rate of change as a derivative (e.g., dP/dt, dv/dt, dD/dp).
- Include a constant of proportionality where the relationship is proportional.
- Use correct signs to reflect increase or decrease (e.g., negative for decay).
- For pure mathematics, eliminate arbitrary constants to form the differential equation.
Examiner Tips
- 💡When asked to differentiate from first principles, always start with the definition and show the limit step clearly.
- 💡For sketching the gradient function, mark key points such as stationary points and where the gradient is positive or negative.
- 💡When using the second derivative to determine concavity, state the sign of f'' and interpret it in terms of the curve's shape.
- 💡Write down the standard derivative you are using before applying it, especially for trigonometric and exponential functions.
- 💡Check each term separately; do not rush and miss constant multiples.
- 💡For expressions with several terms, differentiate term by term and simplify where possible.
- 💡For tangents and normals, clearly state the point and gradient before forming the equation.
- 💡When classifying stationary points, show your method: either second derivative or a sign table.
- 💡For increasing/decreasing, solve the inequality dy/dx > 0 or < 0 and state the intervals clearly.
- 💡Before differentiating, classify the expression: is it a product, a quotient, a composition, or a combination? This determines the first rule to use.
- 💡When using the chain rule with a function inside a function, write down the inner function u and the outer function y in terms of u to avoid missing factors.
- 💡For connected rates, draw a diagram and list all given rates and quantities; then form the chain rule equation linking them.
- 💡For inverse functions, find dx/dy first, then take the reciprocal, and substitute the y-value that corresponds to the given x-value.
- 💡Check your answer by differentiating a simpler similar expression or by using numerical approximation for a specific value.
- 💡For implicit differentiation, write d/dx in front of each term and apply the chain rule to y-terms; then collect dy/dx terms on one side.
- 💡For parametric equations, clearly write dx/dt and dy/dt before dividing; this reduces errors.
- 💡If asked for the gradient at a point, substitute the coordinates or parameter value after finding the general derivative.
- 💡Check whether the relation can be rearranged explicitly; if not, implicit or parametric methods are needed.
- 💡Keep expressions in terms of x and y (or t) as required; do not substitute prematurely unless asked.
- 💡Underline key phrases such as 'rate of change', 'proportional to', and 'with respect to' to identify the derivative and variables.
- 💡Define your symbols clearly at the start of your answer, especially in context questions, to avoid ambiguity.
- 💡Check that your equation makes sense dimensionally and that the signs match the described behaviour.
Common Mistakes
- Confusing the derivative with the original function. Correction: remember f'(x) gives the gradient, not the y-value; for example, if f(x) = x^2, then f'(x) = 2x, which is the gradient at x.
- Thinking that f''(x) = 0 always means a point of inflection. Correction: a point of inflection requires a change in concavity, so check that f'' changes sign; for example, y = x^4 has f''(0) = 0 but no inflection.
- Incorrectly applying first principles by omitting the limit or making algebraic errors. Correction: write the full definition, expand carefully, and take the limit as h tends to 0.
- Forgetting to multiply by the constant k when differentiating e^{kx}, sin kx, etc. Correction: always apply the chain rule factor k.
- Treating ln x as a power of x, e.g. rewriting ln x as x^{-1} and differentiating to -x^{-2}. Correction: d/dx(ln x)=1/x; ln x is not a power function, so the power rule does not apply.
- Omitting the constant k when differentiating tan kx, e.g. writing sec^2 kx instead of k sec^2 kx. Correction: d/dx(tan kx)=k sec^2 kx, so the factor k must be included.
- Using the wrong gradient for the normal: forgetting that the normal gradient is the negative reciprocal of the tangent gradient. Correction: if tangent gradient is m, normal gradient is -1/m.
- Assuming any point where dy/dx = 0 is a maximum or minimum without checking; it could be a point of inflection. Correction: use the second derivative test or check sign change of dy/dx.
- Confusing increasing/decreasing with positive/negative function values rather than gradient. Correction: increasing means dy/dx > 0, decreasing means dy/dx < 0.
- Writing the product rule as u'v' instead of u'v + uv'. Correction: remember the sum of two terms, each with one differentiated factor.
- Reversing the numerator in the quotient rule to u dv/dx − v du/dx. Correction: the correct numerator is v du/dx − u dv/dx, so the derivative of the numerator u (du/dx) appears first with a positive sign, and the derivative of the denominator v (dv/dx) appears second with a negative sign.
- Forgetting to multiply by the inner derivative in the chain rule, e.g. differentiating sin(2x) as cos(2x). Correction: always multiply by the derivative of the inside function, giving 2cos(2x).
- In connected rates, omitting a necessary rate or using the wrong variable. Correction: write the full chain explicitly before substituting.
- For inverse functions, finding dx/dy and forgetting to take the reciprocal: correction, the required derivative is dy/dx = 1/(dx/dy), so after differentiating x with respect to y, take the reciprocal and evaluate at the corresponding point.
- Forgetting to multiply by dy/dx when differentiating a term in y, e.g. writing d/dx(y²) = 2y instead of 2y dy/dx. Correction: always apply the chain rule to y-terms.
- In parametric differentiation, inverting the formula to dy/dx = (dx/dt)/(dy/dt). Correction: dy/dx is (dy/dt) divided by (dx/dt).
- Making algebraic errors when rearranging to make dy/dx the subject, such as dropping negative signs. Correction: work through each step carefully and check by substitution.
- Differentiating a product of x and y incorrectly, e.g. d/dx(xy) = 1 instead of x dy/dx + y. Correction: use the product rule for mixed terms.
- Using the wrong derivative: for example, writing dP/dx instead of dP/dt when time is the independent variable. Always check which variable is changing with respect to which.
- Forgetting the constant of proportionality: if a rate is 'proportional to', a constant k must be included. Omitting it loses the generality of the relationship.
- Incorrect sign: if a quantity decreases, the derivative should be negative. For example, demand decreasing as price increases gives dD/dp = -k, not +k.