Two definitions of total derivative Wikipedia says 
In the mathematical field of differential calculus, a total derivative or full derivative of a function $f$ of several variables, e.g. $t,x,y$, etc., with respect to an exogenous argument, e.g.,$t$, is the limiting ratio of the change in the function's value to the change in the exogenous argument's value (for arbitrarily small changes), taking into account the exogenous argument's direct effect as well as indirect effects via the other arguments of the function. $\frac{df}{dt}=\frac{df}{dx}\frac{dx}{dt}+\frac{df}{dy}\frac{dy}{dt}$
A caltech math note (http://www.math.caltech.edu/~dinakar/08-Ma1cAnalytical-Notes-chap.2.pdf page 2~3) says: total derivative is (Jacobian) such that
$\lim_{h\rightarrow0}\frac{\|f(x+h)-f(x)-L(h)\|}{\|h\|}=0$
Is there any relation between them or are they just two different things?
 A: Suppose that the function
$$f:X\times Y \to Z $$ is Frechet differentiable (2nd definition) with derivative(Jacobian) $d_{\text{Frec},\mathbf x_0}f$ at some point $\mathbf x_0=(x_0,y_0)$. Suppose also we have that the differentiable functions
$$ x:[0,T] \to X,\quad y:[0,T] \to Y, \quad \mathbf x(t)=(x(t),y(t))^T\in X\times Y, \quad \mathbf x(0) = \mathbf x_0.$$ 
are functions of some parameter $t\in [0,T]$. 
Then the total derivative(first definition) $d_{\text{total},\mathbf x_0}f$ at $\mathbf x_0$ wrt the parameter $t$ can be computed as
$$d_{\text{total},\mathbf x_0}f =  \left.\frac{d}{dt} f(x(t),y(t)) \right|_{t=0}\overset{\substack{chain\\rule}}{=} (d_{\text{Frec},\mathbf x_0})f\cdot \mathbf x'(0), $$
where the $\cdot$ is matrix multiplication. Suppose that $X=Y=Z=\mathbb R$. Then $d_{\text{Frec},\mathbf x_0}f = (\partial_xf(\mathbf x_0), \partial_y f(\mathbf x_0)) $ is a $1\times 2$ matrix, and $\mathbf x'=(x',y')^T$ is a $2\times 1$ matrix. One checks that this gives 
$$ d_{\text{total},\mathbf x_0}f = \partial_xf(\mathbf x_0) x'(0) + \partial_yf(\mathbf x_0) y'(0) $$
as expected. A similar statement holds with block matrices if instead $\dim X = \dim Y = \dim Z > 1$.
