By A. F. Bermant
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Extra info for A Course of Mathematical Analysis, Part II
Example. y' = -au = 1,' 1" = sin x v. Then y ~~t'. We have: ay v' = vu,,-l --. 3:. n x)sinx ucos x) (_1_x + cos cosxln tanx). A special method-"logarithmic differentiation" -was recommended earlier for finding the derivative of such a function. :; + av dx as dw + ... + aw dx . e. e. on the assumption that the remaining arguments remain constant during differentiation in spite of the fact that they depend on x. For example, if azla then III. Simple rules analogous to those for the case of one variable (Sec.
T) if it has a differential at this point. REMARK. If du = 0, u is a constant. For, it follows from the au identity :Jdx ux au + :uyJ dy + ... e. that u is independent of x, y, Z, •.. au = 0, :J uy ... e. is constant. 146. Geometrical Interpretation of the Differential. Just as the deri- vative and differential of a function of one variable are connected with the tangent to a curve-the graph of the function-, the derivatives and differential of a function of two variables are connected with the tangent plane to a surface-the graph in this case.
In the particular case when all the arguments u, v, ... , ware functions of one independent variable x, we are in fact concerned with a function of x only. The (ordinary) derivative of the function-which is termed in this case the total derivative-now has' the form dz az du as dv az dw -dx = ---+ ... + au d;"lJ + av dx aw -dx. (***) Example. y' = -au = 1,' 1" = sin x v. Then y ~~t'. We have: ay v' = vu,,-l --. 3:. n x)sinx ucos x) (_1_x + cos cosxln tanx). A special method-"logarithmic differentiation" -was recommended earlier for finding the derivative of such a function.
A Course of Mathematical Analysis, Part II by A. F. Bermant