| d(u + v + ... + z) | = | du | + | dv | + ... + | dz | . |
| dx | dx | dx | dx |
(iii.) For a product uv
| d(uv) | = u | dv | + v | du | . |
| dx | dx | dx |
(iv.) For a quotient u/v
| d(u/v) | = ( v | du | − u | dv | ) / v2. |
| dx | dx | dx |
(v.) For a function of a function, that is to say, for a function y expressed in terms of a variable z, which is itself expressed as a function of x,
| dy | = | dy | · | dz | . |
| dx | dz | dx |
In addition to these formal rules we have particular results as to the differentiation of simple functions. The most important results are written down in the following table:—
| y | dy/dx |
| xn | nxn−1 for all values of n |
| logax | x-1 logae |
| ax | ax logea |
| sin x | cos x |
| cos x | −sin x |
| sin−1x | (1 − x2)−1/2 |
| tan−1x | (1 + x2)−1 |
Each of the formal rules, and each of the particular results in the table, is a theorem of the differential calculus. All functions (or rather expressions) which can be made up from those in the table by a finite number of operations of addition, subtraction, multiplication or division can be differentiated by the formal rules. All such functions are called explicit functions. In addition to these we have implicit functions, or such as are determined by an equation containing two variables when the equation cannot be solved so as to exhibit the one variable expressed in terms of the other. We have also functions of several variables. Further, since the derived function of a given function is itself a function, we may seek to differentiate it, and thus there arise the second and higher differential coefficients. We postpone for the present the problems of differential calculus which arise from these considerations. Again, we may have explicit functions which are expressed as the results of limiting operations, or by the limits of the results obtained by performing an infinite number of algebraic operations upon the simple functions. For the problem of differentiating such functions reference may be made to [Function].