Definition:Derivative

= Derivative at a Point =

Real Functions
Let $$I$$ be an open real interval.

Let $$f: I \to \R$$ be a real function defined on $$I$$.

Let $$\xi \in I$$ be a point in $$I$$.

Let $$f$$ be differentiable at the point $\xi$.

That is, suppose the limit $$\lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$$ exists.

Then this limit is called the derivative of $$f$$ at the point $$\xi$$ and is variously denoted:
 * $$f^{\prime} \left({\xi}\right)$$;
 * $$D f \left({\xi}\right)$$;
 * $$D_x f \left({\xi}\right)$$;
 * $$\frac{d}{dx} \left({\xi}\right)$$.

Alternatively it may be written:
 * $$f^{\prime} \left({\xi}\right) = \lim_{h \to 0} \frac {f \left({\xi + h}\right) - f \left({\xi}\right)} {h}$$

Derivative on an Interval
Let $$f$$ be a real function defined on an open interval $$I$$.

Let $$f$$ be differentiable on the interval $I$.

Then $$f^{\prime}: I \to \R$$ is defined as the real function whose value at each point $$x \in I$$ is $$f^{\prime} \left({x}\right)$$.

It can be variously denoted as:
 * $$\frac{df}{dx}$$;
 * $$\frac{d}{dx} \left({f}\right)$$;
 * $$f^{\prime} \left({x}\right)$$;
 * $$D f \left({x}\right)$$.
 * $$D_x f \left({x}\right)$$.

With Respect To
Let $$f$$ be a real function which is differentiable on an open interval $$I$$.

Let $$f$$ be defined as an equation: $$y = f \left({x}\right)$$.

Then the derivative of $$y$$ with respect to $$x$$ is defined as $$y^{\prime} = \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} {h} = D_x f \left({x}\right)$$.

This is frequently abbreviated as derivative of $$y$$ WRT $$x$$.

We introduce the quantity $$\delta y = f \left({x + \delta x}\right) - f \left({x}\right)$$.

This is often referred to as the small change in $$y$$ consequent on the small change in $$x$$.

Hence the motivation behind the popular and commonly-seen notation:
 * $$\frac{dy}{dx} \ \stackrel {\mathbf {def}} {=\!=} \ \lim_{\delta x \to 0} \frac {f \left({x + \delta x}\right) - f \left({x}\right)} {\delta x} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x}$$

Hence the notation $$f^{\prime} \left({x}\right) = \frac{dy}{dx}$$. This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes.

However, it has the disadvantage that the variable $$x$$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done. For practical applications, however, this is not usually a problem.

Complex Functions
The definition for a complex function is similar to that for real functions.

Let $$f \left({z}\right): \C \to \C \ $$ be a single-valued continuous complex function in a domain $$D \subseteq \C$$.

Let $$z_0 \in D$$ be a point in $$D$$.

Let $$f$$ be differentiable at the point $z_0$.

That is, suppose the limit $$\lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} {h}$$ exists as a finite number and is independent of how the complex increment $$h \ $$ tends to $$0 \ $$.

Then this limit is called the derivative of $$f$$ at the point $$z_0$$ and is variously denoted:
 * $$f^{\prime} \left({z_0}\right)$$;
 * $$D f \left({z_0}\right)$$;
 * $$D_z f \left({z_0}\right)$$;
 * $$\frac{d}{dz} \left({z_0}\right)$$.

Further, let $$f$$ be differentiable in $D \ $.

Then $$f^{\prime}: D \to \C$$ is defined as the complex function whose value at each point $$z \in D$$ is $$f^{\prime} \left({z}\right)$$.

It can be variously denoted as:
 * $$\frac{df}{dz}$$;
 * $$\frac{d}{dz} \left({f}\right)$$;
 * $$f^{\prime} \left({z}\right)$$;
 * $$D f \left({z}\right)$$.
 * $$D_z f \left({z}\right)$$.

= Second Derivative =

Let $$f$$ be a real function which is differentiable on an open interval $$I$$.

Hence $$f^{\prime}$$ is defined as above.

Let $$f^{\prime}$$ be differentiable on the interval $I$.

Let $$\xi \in I$$ be a point in $$I$$.

Let $$f^{\prime}$$ be differentiable at the point $\xi$.

Then the second derivative $$f^{\prime \prime} \left({\xi}\right)$$ is defined as $$\lim_{x \to \xi} \frac {f^{\prime} \left({x}\right) - f^{\prime} \left({\xi}\right)} {x - \xi}$$.

Again, it is variously denoted:
 * $$f^{\prime \prime} \left({\xi}\right)$$;
 * $$D^2 f \left({\xi}\right)$$;
 * $$D_{xx} f \left({\xi}\right)$$;
 * $$\frac{d^2}{dx^2} \left({\xi}\right)$$.

And again, it may alternatively be written:
 * $$f^{\prime \prime} \left({\xi}\right) = \lim_{h \to 0} \frac {f^{\prime} \left({\xi + h}\right) - f^{\prime} \left({\xi}\right)} {h}$$

Thus the second derivative is defined as the derivative of the derivative (which, in this context, can be referred to as the "first derivative").

If $$y = f \left({x}\right)$$, then it can also denoted by $$y''$$ or $$\frac{d^2y}{dx^2}$$.

If $$f^{\prime}$$ is differentiable, then it is said that $$f$$ is doubly differentiable, or twice differentiable.

Higher Derivatives
Higher derivatives are defined in similar ways.

In general, the notation for the $$n$$th derivative at a point $$\xi$$ is given by:
 * $$f^{\left({n}\right)} \left({\xi}\right)$$;
 * $$D^n f \left({\xi}\right)$$;
 * $$D_{x \left({n}\right)} f \left({\xi}\right)$$;
 * $$\frac{d^n}{dx^n} \left({\xi}\right)$$.

If the $$n$$th derivative exists for a function, then $$f$$ is described as being $$n$$ times differentiable.

Order of a Derivative
The order of a derivative is the number of times it has been derived.

For example:
 * a first derivative is of first order, or order $$1$$;
 * a second derivative is of second order, or order $$2$$,

and so on.

Differentiation
The process of obtaining the derivative of a function $$f$$ with respect to $$x$$ is known as differentiation (of $$f$$) with respect to $$x$$, or differentiation WRT $$x$$