Contents

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Alternative Title: The derivative

Derivative, the measure of the rate at which the value y of a function f changes with respect to x as Δx, the distance between two points at the x-axis, becomes infinitely small. It can be defined using the following formula:

$$ \lim\limits_{\Delta x\to 0}{\Delta y\over\Delta x}\space =\space \lim\limits_{\Delta x\to 0}{f(x+\Delta x) - f(x)\over\Delta x} $$

The derivative of functions

Notation

$$ \begin{aligned}{\rm Leibniz'\space notation:}\space & dy\over dx\\[15pt] {\rm Lagrange's \space notation:}\space & f'(x) \end{aligned} $$

The chain rule states that the derivative of a composite function is determined by:

$$ {d\over dx}\lbrack f(g(x))\rbrack =f'(g(x))\cdot g'(x) $$

Derivative of **vector functions**

$$ r'(t)=\lim_{\Delta t\to 0}{\vec r(t+\Delta t)-\vec r(t)\over\Delta t} $$

$$ {\rm if}\space \vec r(t)=[x(t),y(t)]{\rm ,\space then}\space \vec r\space '(t)=[x'(t),y'(t)] $$

Functions for velocity (v(t)) and acceleration (a(t)):

$$ \vec v(t)={d\over dt}\vec r(t) $$

$$ \vec a(t)={d^2\over dt^2}\vec r(t) $$

Rules for calculation

General formula

$$ {d^a\over dx^a}(k\cdot x^n+C)=k\cdot n(n-1)\ldots (n-a+1)\cdot x^{n-a}\cdot x' $$

Note that x can represent either a variable (algebraic x), constant (k_2) or a function (f(x)).

From this we can conclude that:

$$ \def\arraystretch{0.25} \begin{array}{c}\begin{aligned}(k\cdot \sqrt[t]{x^n}+C)'=k\cdot {n\over t}x^{{n\over t}-1}\cdot x'\end{aligned}\\\\\hline\\\hline\end{array} $$

Addition

$$ (u(x)+v(x))'=u'(x)+v'(x) $$

Subtraction

$$ (u(x)-v(x))'=u'(x)-v'(x) $$

Product and quotient

$$ \begin{aligned}(u\cdot v)' & =u'v+uv'\\[5pt] ({u\over v})' & ={u'v-uv'\over v^2}\end{aligned} $$

Involvement of the natural logarithm

$$ \begin{aligned}(a^x)' & =a^x\cdot ln\space a\\[6pt] {d\over dx} \lbrack ln\space (kx^p)\rbrack & ={p\over x}\cdot x'\end{aligned} $$

Derivative of f'(x)

The point at which the derivative of a continuous function f is zero, is:

$$ \begin{aligned} {\rm a\space maxima\space if}\space & f''(a)<0\\ {\rm a\space minima\space if}\space & f''(a)>0 &\end{aligned} $$

In the interval [a, b], the graph of a continuous function f is:

$$ \begin{aligned} {\rm concave\space if}\space & f''(x)<0\\ {\rm convex\space if}\space & f''(x)>0 &\end{aligned} $$

A visual representation of the concept of derivative, using a tangent on a two-dimensional graph. Note that the function for average growth between x + Δx gets increasingly accurate when Δx reaches zero. Photo: Vonvikken - Own work, CC BY-SA 3.0, Wikimedia.