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Differential calculus has widespread applications in ma,y areas (differential equations, statistics. machine learning and many others).

Derivatives

Let us start with some of the basic notation, concepts, and results of elementary and multivariable differential calculus. Throughout this section, we will assume differentiability or multiple differentiability of the functions we discuss. For more details on the conditions for differentiability, see our tutorial. If f is a real-valued function of one variable, x, then its derivative at x, if it exists, is given by
\[ f^{(1)} (x) = f' (x) = \frac{{\text d}f(x)}{{\text d}x} = \lim_{\varepsilon \to 0} \frac{f(x+ \varepsilon ) - f(x)}{\varepsilon} \]
Equivalently, f′(x) is the quantity that gives the first-order Taylor formula for f(x + ε). In other words
\[ f (x + \varepsilon ) = f(x) + \varepsilon\, f' (x) + r_1 (x, \varepsilon ) , \]
where the remainder r₁(x, ε) is a function of x and ε satisfying
\[ \lim_{\varepsilon\to 0} \frac{r_1 (x, \varepsilon )}{\varepsilon} = 0 . \]
The quantity
\[ {\text d}_{\varepsilon}f(x) = \varepsilon\, f' (x) \]
is called the first differential of f at x with increment ε. This increment ε is the differential of x. Later we will use dx in place of ε, that is, write f(x + dx) instead of f(x + ε), to emphasize the fact that u is the differential of x. For notational convenience, we will often denote the differential given in (9.2) simply by df. Generalizations of the above equation can be obtained by taking higher ordered derivatives; that is, with the ith derivative of f at x defined as
\[ \texttt{D}^k f(x) = \lim_{\varepsilon\to 0} \frac{f^{(k-1)} (x + \varepsilon ) - f^{(k-1)} (x)}{\varepsilon} \qquad (k=1,2,\ldots ) , \]
we have the kth-order Taylor formula
\begin{align*} f(x + \varepsilon ) &= f(x) + \sum_{k=1}^m \frac{\varepsilon^k}{k!}\, f^{(k)} (x) + r_m (x, \varepsilon ) \\ &= f(x) + \sum_{k=1}^m \frac{{\text d}^k_{\varepsilon} f(x)}{k!} + r_m (x, \varepsilon ) , \end{align*}
where rm(x, ε) is a function of ε and x satisfying
\[ \lim_{\varepsilon\to 0} \frac{r_m (x, \varepsilon )}{\varepsilon^m} = 0 , \]
and
\[ {\text d}_{\varepsilon}^k f(x) = \varepsilon^k f^{(k)} (x) , \]
or simply dkf, is the kth differential of f at x with increment ε.

The chain rule is a useful formula for calculating the derivative of a composite function. If y, g, and f are functions such that y = g(f(x)), then

\[ y' (x) = g' (f(x))\, f' (x) . \]
If f is a real-valued function of the n × 1 vector x = [x₁, … , xn]′, then its derivative at x, if it exists, is given by the 1 × n row vector
\[ \frac{\partial}{\partial x'} \,f(x) = \left[ \frac{\partial}{\partial x_1}\,f(\mathbf{x}) , \cdots , \frac{\partial}{\partial x_n}\,f(\mathbf{x}) \right] , \]
where    

 

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