Differential calculus
Differential calculus for IB Analysis and Approaches HL: derivative rules, tangents, extrema, optimisation, rates of change and series, with IB exam questions.
The derivative measures how a function changes: its value at a point is the gradient of the tangent and the instantaneous rate of change. In this topic you learn to calculate it (differentiation rules, chain rule, inverse functions, exponential and logarithmic functions, implicit differentiation) and to use it to study a function: increase and decrease, extrema, concavity and points of inflexion. With it you solve optimisation, rate of change and motion along a line problems, and you evaluate limits with L'Hôpital's rule or approximate functions with Maclaurin series. It appears in all Higher Level papers, with and without a GDC.
Recommended order
Work through the sections in order: derivative at a point, differentiability and continuity, calculating derivatives, chain rule, inverse function, exponentials and logarithms, increase and decrease and extrema, optimisation, rate of change, concavity, Rolle's and mean value theorems and L'Hôpital, Taylor and Maclaurin, rectilinear motion and implicit differentiation. At the end you will find the IB exam questions grouped by type. Practise the rules until you differentiate without thinking: it is the basis of integral calculus (topic 11).
Functions, limits and continuity (topic 9).
Sections in this topic
Derivative of a function at a point
The derivative at a point: rate of change, limit definition, tangent and normal lines and one-sided derivatives. IB Analysis and Approaches HL.
3 worked examples Section 2Differentiability and continuity; operations with derivatives
Differentiable implies continuous, corners and vertical tangents, piecewise functions with parameters, and the product and quotient rules. IB AA HL.
3 worked examples Section 3Calculating derivatives
Derivatives of powers, roots, polynomials, sine, cosine and tangent, and how to combine them with the product and quotient rules. IB AA HL.
3 worked examples Section 4Chain rule
How to differentiate composite functions with the chain rule, in function and Leibniz notation, with several layers, and higher-order derivatives. IB AA HL.
3 worked examples Section 5Derivative of the inverse function and of the inverse trigonometric functions
Differentiate an inverse function without solving for it, and find the derivatives of arcsin, arccos and arctan. IB Analysis and Approaches HL.
3 worked examples Section 6Derivatives of exponential and logarithmic functions
Derivatives of e^x, a^x, ln x and log_a x, logarithmic differentiation of f(x)^g(x) and a summary table with the chain rule. IB Analysis and Approaches HL.
3 worked examples Section 7Increasing, decreasing and extrema
Use the sign of the derivative to find where a function increases or decreases, its local extrema and its absolute extrema. IB Analysis and Approaches HL.
3 worked examples Section 8Optimisation: maxima and minima in context
Solve optimisation problems with derivatives: set up the function, reduce it to one variable and justify the extremum. IB Analysis and Approaches HL.
3 worked examples Section 9Rate of change problems
Solve related rates problems, where quantities change over time, by differentiating with the chain rule. IB Analysis and Approaches HL.
3 worked examples Section 10Concavity and points of inflexion
Use the second derivative to find concavity and points of inflexion, classify extrema and sketch a function. IB Analysis and Approaches HL.
3 worked examples Section 11Rolle's and mean value theorems; L'Hôpital's rule
Rolle's theorem, the mean value theorem and L'Hôpital's rule for indeterminate limits, with worked examples. IB Analysis and Approaches HL.
3 worked examples Section 12Taylor and Maclaurin expansions
Taylor and Maclaurin polynomials and their series for e^x, sin x, cos x and ln(1+x), used for approximations and limits. IB Analysis and Approaches HL.
3 worked examples Section 13Rectilinear motion: position, velocity and acceleration
Using the derivative to study straight-line motion: velocity, acceleration, speed, changes of direction and distance travelled. IB Analysis and Approaches HL.
3 worked examples Section 14Implicit differentiation
Implicit differentiation step by step: tangents, normals, horizontal tangents and second derivatives without solving for y. IB Analysis and Approaches HL.
3 worked examples End of topicIB exam questions on differential calculus
IB exam questions on differential calculus: derivatives, tangents, optimisation, rates of change, L'Hôpital's rule and Maclaurin series, with worked solutions.
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