Narayanadash. - Ordinary and Partial Differentiation, Mean Value Theorems, Taylors Series, Orthogonal Coordinates

Ifthe physical quantity represents electrical potential difference, itsrate of change with respect to distance in a particular direction would represent electric field strength in that direction. In this manner we can relate physical quantities like distance and acceleration which are different from each other but depend upon each other. Consider simply two hills OPQ and OPQ in the figure above . We have to compare slopes (slant-ness or steepness) of the two hills; A logical thinking would suggest that the hill is more steep at a point Z where climb more distance upwards ZR than where we have to climb less distance upwards ZR while advancing the same distance ZT = ZT towards centre of the mountains .It is not important to measure how much we climb up, rather how much we have to climb up while advancing unit distance towards centre of the hill.This ratio RT/ZT in case of graph of f(x) or RT/Z/T can be taken as a measurement of slopes. This may be observed to be tan A or tan B in the two cases . tan A > tan B consequently the first graph is more steep than the second.

Please note that when the point P is reached, the hills start sloping downwards i.e., the slopes become negative. and consequently the function, here the height above ground level starts decreasing. Because the hills were taken as straight, slope in one side remainedconstant from bottom to the top. Had the contour of the hill been like the curve OPQU as in the above figure ,we should consider a small distance PQ and measure the slope as QR/PR at the point P, for the slope is changing from point to point, the hill is more steep at some points and less at others. But how small an arc PQ is required for this purpose of measuring the slope ? The nearer Q is to P the better would be our approximation. We may afford to be exact theoretically, if we take ZPQV to be tangent to the curve instead of secant.

This ratio or slope is called differential co-efficient of the curve or of the function f(x) at point P (x=a) and denoted by symbol , or at x = a, written as f(x) x=a = = = = tan (1), being the angle of the tangent of the curve at the point (x, f(x)) with the X-axis called the slope angle of the tangent QS of the curve at the point.. Note that the angle QPR = angle PSX = . We mean y as change in y, x as change in x, f as change inf(x), or, f(x+ x) - f(x). Further,note in the figure 3.1.a, that the slope of the line PO is +ve and that of PQ is ve, hence different. dy/dx being the limit of the slopes this limit here does not exist, as the left handed limit is different from the right handed limit. So derivative at a sharp point on a smooth curve does not exist.

It is not the ratio of dy and dx ; but the limit of the ratio y and x. (Note that the tangent makes the same angle with X-axis as it makes with the horizontal line at P. Note here that f(a) has to exist in thefirst place at x = a if we are to calculate f(x)x=a this means or ;This is the criteria for the function to be continuous. So the function