# Total derivative?

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?

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.

• 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? Sep 15, 2019 at 18:58
• Derivatives are used for linear approximation near the given point to avoid complicated evaluations. Sep 15, 2019 at 19:18

The total differential is intuitevely, straight-forwardly the Sum of ALL differentials.

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