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Please explain me the difference between $\lim_{x->0}\frac{\partial E}{\partial x}$ and $\lim_{x->0}dE/dx$.In physics I encountered something similar while reading about Newton's Law of Fluids.While in F.M. White its done using partial derivatives.I want to know the physical difference instead of the highly mathematical one. Having said that I am well conversed with the first principle of derivatives.

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  • $\begingroup$ So, what do you understand of derivatives? $\endgroup$ – Solar Mike Aug 20 '17 at 18:09
  • $\begingroup$ That is is basically the instantaneous value of a function.The secants dawn between any two points basically tend to a tangent which is the derivative.I have read that for tarasov and am pretty sure about this concept. $\endgroup$ – gateprep Aug 20 '17 at 18:55
  • $\begingroup$ So, a derivative is based on all the variables changing, a partial is some or only one but not all... $\endgroup$ – Solar Mike Aug 20 '17 at 19:37
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For a detailed explanation search WikiPedia for derivative, partial derivative and total derivative. For a short, non-mathematical, summary see below.

The partial derivative of a function of several variables is it's derivative with respect to one of those variables, assuming that all other variables are constant. The total derivative does not make this assumption and includes all indirect dependencies to find the overall dependency with respect to the variable of interest.

As an example, consider $\frac{df}{dx}$, the total derivative of the function $f(x,y)$ with respect to the variable $x$:

$$\frac{\operatorname df}{\operatorname dx}=\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{\operatorname dy}{\operatorname dx}$$

which depends on both $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ (i.e. the partial derivatives of $f$ with respect to $x$ and $y$).

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Consider a surface S in 3-space and its gradient

$ {\hat{i} }\frac{\partial{S}}{\partial{x}} + {\hat{j} }\frac{\partial{S}}{\partial{y}} $ . Then each partial represents the slope along the given axis, as though you'd taken a 2-dimensional slice of the surface and looked at the slopeof the line that resulted.
If you then find the direction (vector in x and y) which maximizes the gradient, you'll find the path of steepest descent.

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