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For a differentiable function of one variable, $y = f(x)$, we define the differential $dx$ to be an independent variable; that is, $dx$ can be given the value of any real number. The differential of $y$ is then defined as $$ dy = f'(x)dx $$ To me, this makes sense because the definition of a derivative - that $f'(x) = \frac{dy}{dx}$. Geometrically, this figure also helps a lot in understanding the relations, particularly between $\Delta y$ and $dy$: figure 1

Now the total derivative as (with $dx$ and $dy$ as independent variables) $$ dz = f_x(x,y)dx+f_y(x,y)dy $$ If we let $\Delta x = x - a$ and $\Delta y = y - b$, the total derivative can be used in the linear approximation of a function like $f(x,y) \approx f(a,b) + dz$.

The book that I am reading, Calculus Early Trancendentals 8th edition, then tells me that $dz$ represents the change in height of the tangent plane, whereas $\Delta z$ represents the change in height of the surface $z = f(x,y)$ when $(x,y)$ changes from $(a,b)$ to $(a + \Delta x, b + \Delta y)$ and shows the figure figure 2

The questions: So I simply do not understand why the total derivative is defined as it is - where does the formula come from? What is the logic behind it? Neither do I understand figure 2 in relation to the formula; why does $f_x(x,y)dx + f_y(x,y)dy$ give the change in height of the tangent plane between two points?

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The tangent plane at a given point is a linear function passing through the point $(x_o,y_o,f(x_o,y_o))$ and has the same partial derivatives as $f(x,y)$ at the point of tangency.

Thus the equation of tangent plane is

$$z=f(x_o,y_o)+\frac {\partial {f}}{\partial x}( x-x_o)+\frac {\partial {f}}{\partial {y}}(y-y_o) $$

Where partial derivatives are taken at the point of tangency.

That explains the total linear change of the function at a given point.

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  • $\begingroup$ Thanks! But how does this reflect the that $dz$ represents the change in height of the tangent plane, whereas $\Delta z$ represents the change in height of the surface $z = f(x,y)$? And what can this "total derivative" be used for other than a linear approximation - if anything? $\endgroup$
    – That Guy
    Sep 15, 2019 at 18:58
  • $\begingroup$ Derivatives are used for linear approximation near the given point to avoid complicated evaluations. $\endgroup$ Sep 15, 2019 at 19:18
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The total differential is intuitevely, straight-forwardly the Sum of ALL differentials.

I can't understand what you yourself doesn't understand.

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