differential and derivatives

Initially, differentials were conceived as infinitesimal changes. The differential \[dy\] of a function \[y\] was thought to be an infinitesimal change in \[y\], and the derivative \[\frac{dy}{dx}\] was the ratio of the infinitesimal change in \[y\] to that in \[x\]. Similarly, integrals were viewed as the sum of areas of infinitely many infinitesimally thin rectangles under a curve. Each rectangle had a height \[y\] and a base \[dx\], so its area was \[y\cdot dx\]. Summing these areas gave the integral, symbolized by \[\int\].

However, infinitesimals were not rigorously defined, leading to logical inconsistencies. For instance, in the product rule for differentiation, terms involving products of differentials like \[df\cdot dg\] were often dismissed as negligible without a solid justification. To address these issues, calculus was redefined using limits. The derivative is now understood as the limit of the average rate of change as the interval approaches zero: \[\frac{dy}{dx}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}\]

Integrals are defined as the limit of Riemann sums, which are finite sums of areas of rectangles approximating the area under a curve: \[\int^{b}_{a}f(x)dx=\lim_{n\to\infty}\sum^{n}_{i=1}f(x_{i})\Delta x\].

Despite the shift to limits, the notation involving differentials like \[dx\] and \[dy\] persisted because it is convenient and suggestive. In integration, \[dx\] acts like a "closing parenthesis" indicating the variable of integration. To give differentials a rigorous meaning, they are defined in terms of derivatives. The differential \[dy\] is approximated by the change along the tangent to the curve: \[dy\approx y^{\prime}(a)\Delta x\], and in the limit \[\Delta x\to0\], \[dy=y^{\prime}(a)dx\].

The distinction between derivatives and differentials is crucial. Derivatives measure rates of change and are dimensionally ratios, such as velocity being distance over time: \[\frac{dy}{dx}\] = rate of change of \[y\] with respect to \[x\]. In contrast, differentials measure actual changes and have the same units as the quantity being measured, like distinguishing between the rate of inflation and the actual change in price: \[dy\] = actual change in \[y\].

The distinction between differentials and derivatives becomes even more apparent in higher dimensions, such as in multivariable calculus. For instance, in multivariable functions, differentials can help in understanding how changes in multiple variables affect the function's value: \[dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy\]. Here, the partial derivatives \[\frac{\partial z}{\partial x}\] and \[\frac{\partial z}{\partial y}\] measure the rates of change of \[z\] with respect to each variable independently, while differentials \[dx\] and \[dy\] represent the infinitesimal changes in the variables \[x\] and \[y\].

Essentially, derivatives tell you how fast one variable changes in relation to another, while differentials represent small changes in variables and can be used to approximate how much a function changes due to small changes in its inputs.