Differentiation rules, applications of derivatives, integration techniques and differential equations, as covered in Class 11 and 12 NCERT Mathematics and tested in JEE Main and Advanced. A sensible order is the differentiation rules first, then their applications (tangents, maxima and minima, rates of change), then integration.
21 reference pages
The chain rule in function and Leibniz notation, a table of standard chain-rule derivatives, four worked examples and common mistakes.
Definite integrals and the fundamental theorem of calculus, the properties P0 to P7 in a table, worked examples using them, and common mistakes.
The power rule for whole, negative and fractional exponents, with a table of examples, two worked examples, a graph of x³ and its derivative, and common mistakes.
The product rule for two and three functions, logarithmic differentiation, worked examples and common mistakes.
The quotient rule with three worked examples, including the derivative of tan x, and the common mistakes such as getting the numerator's order wrong.
Derivatives as rates of change: a summary of every application, worked examples on area, velocity and related rates, and common mistakes.
Implicit differentiation step by step, with worked examples on product terms, second derivatives and an inverse trig identity, plus common mistakes.
Finding where a function increases or decreases from the sign of its derivative, with a sign chart, a graph, worked examples and common mistakes.
The first and second derivative tests, absolute extremes on a closed interval, an optimisation problem, a graph and common mistakes.
Rolle's theorem and the mean value theorem: conditions, graphs, a side-by-side comparison, worked examples and when the theorem fails.
Slopes and equations of tangent and normal lines to a curve, special cases, four worked examples, a graph and common mistakes.
Integration by partial fractions with a table of forms for distinct, repeated and quadratic factors, three worked examples and common mistakes.
All nine standard integrals of particular functions in one table, an example of each, completing the square, and common mistakes.
Integrating trig functions with power-reduction and product-to-sum identities: a table of identities, four worked examples and common mistakes.
The shortcut for integrating e^x[f(x) + f'(x)], why it works, a table for spotting f, worked examples and common mistakes.
Finding the area bounded by a curve or between two curves with definite integrals, with shaded graphs, worked examples and common mistakes.
Order and degree, the separable, homogeneous and linear types with a worked example of each, forming equations, and common mistakes.
Integration by parts with the ILATE rule, worked examples on logarithms, repeated parts and the cyclic e^x sin x case, and common mistakes.
The three-part continuity check, types of discontinuity with a graph, continuity versus differentiability, worked examples and common mistakes.
Approximating values with differentials, why dy approximates Δy (with a graph), worked examples on roots, volume and error, and common mistakes.
Linear programming step by step: forming constraints from a word problem, the feasible region graph, the corner point method and common mistakes.