Total derivative

A total derivative accounts for every dependence of a quantity on the variable of differentiation. For a differentiable function $F(t,x_1,\dots ,x_n)$ evaluated along differentiable coordinate functions $x_i(t)$, the multivariable chain rule givesAll the partial derivatives on the right are evaluated at $(t,x_1(t),\dots ,x_n(t))$. The first term captures explicit dependence on $t$; the remaining terms capture changes transmitted through the coordinates.

For example, take $F(t,x)=t{x}^2$ and $x(t)=t$. The partial derivative with respect to $t$, holding $x$ fixed, is $x^2$, whose value on the path is $t^2$. The total derivative is insteadAt $t=2$, the partial derivative is $4$ and the total derivative is $12$. The difference is the contribution from changing $x$.

If the composite function is continuously differentiable on $[a,b]$, the fundamental theorem of calculus givesThis is the sense in which an integrated total derivative contributes endpoint terms. The endpoint statement concerns the whole composite function along the path. ^[ref].

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:and in 2D: