On the different orders of infinitely small quantities. Infinitely small quantities of a certain order may be neglected in respect of those of an inferior order. The infinitely small increment of a function is in general of the same order as the corresponding increment of the variable, that is to say, their ratio has a finite limit.
Definitions of the derivative and differential of a function of a single variable. Tangents and normals to plane curves, whose equation in linear or polar coordinates is given.
A function is increasing or decreasing, according as its derivative is positive or negative. If the derivative is zero for all values of the variable, the function is constant. Concavity and convexity of curves; points of inflection.
Principle of function of functions. Differentiation of inverse functions.
Differentials of the sums, products, quotients, and powers of functions, whose differentials are known. General theorem for the differentiation of functions composed of several functions.
Differentials of exponential and logarithmic functions.
Differentials of direct and inverse circular functions.
Differentiation of implicit functions.
Tangents to curves of double curvature. Normal plane.
Differential of the area and arc of a plane curve, in terms of rectilinear and polar co-ordinates.