Here, a was the coefficient of the first-order term in the linear approximation of h. In this sense, a is referred to as the derivative “coefficient” of f at x. By slightly transforming the above equation, we can see that the derivative coefficient of f at x satisfies the equation for a.
h→0limhf(x+h)−f(x)−ah=h→0limhr(h)=0
This forms the basis for defining the derivative of a multivariable vector function.
Definition
Let’s denote E⊂Rn as an open set, and x∈E accordingly. For f:E→Rm, if there exists a linear transformation A∈L(Rn,Rm) for h∈Rn that satisfies the following, then f is differentiable at x. Furthermore, A is called the total derivative or simply the derivative of f and is denoted by f′(x).
∣h∣→0lim∣h∣∣f(x+h)−f(x)−A(h)∣=0
If f is differentiable at all points in E, then f is said to be differentiable in E.
Explanation
The term “total” implies entirety, contrasting with partial derivatives. It’s not the total ˇ function, but the total ˇ derivative.
It is important to note that f′(x) is not a function value but a linear transformation satisfying f′(x):E⊂Rn→Rm. Therefore, f′(x)=A can be represented as a matrix as follows.
Then, the total derivative of f, f′, can be seen as a function mapping a certain matrix A every time x∈E⊂Rn is provided. This matrix can be easily obtained from the partial derivatives of f and is also known as the Jacobian matrix.
The total derivative is the ultimate generalization of differentiation for functions defined on finite dimensions, extending the domain and range of f to Banach spaces as the Fréchet derivative. The properties that held for univariate functions naturally hold as well.
Uniqueness
Chain Rule
Theorem
Uniqueness
Let E,x,f be as defined in the Definition. If A1,A2 satisfies (2), then the two linear transformations are equal.
A1=A2
Proof
Let’s denote B=A1−A2. Then, by the triangle inequality, the following holds.
Since h=0, for the above equation to hold, it must be B=0. Thus, we obtain the following.
B=A1−A2=0⟹A1=A2
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Chain Rule
As defined, let’s consider E⊂Rn as an open set and f:E→Rm as a function differentiable in x0∈E. Let g:f(E)→Rk be a function differentiable in f(x0)∈f(E). Also, let’s consider F:E→Rk as the composition of f and g.
F(x)=g(f(x))
Then, F is differentiable in x0, and the total derivative is as follows.