Fundamental Mathematics/Arithmetic/Arithmetic Operation/Differentiation

Differentiation Definition

Differentiation is a mathematics operation to find Gradient of a function (ie. to find slope of the function) . The derivative of a function of a real variable measures the sensitivity to change of the function (output) value with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus denoted as

{\frac {d}{dt}}f(t)=f^{'}(t)

The mathematical operation

{\frac {d}{dt}}f(t)=f^{'}(t)=\lim _{\Delta t\to 0}\Sigma {\frac {f(x+\Delta x)-f(x)}{\Delta x}}

Variation in notation

Leibniz's notation

In Leibniz's notation, an infinitesimal change in $x$ is denoted by $dx$ , and the derivative of $y$ with respect to $x$ is written

{\frac {d}{dx}}y\,\!

Lagrange's notation

In Lagrange's notation, the derivative with respect to $x$ of a function $f (x)$ is denoted $f' (x)$ (read as "f prime of x") or $f x' (x)$ (read as "f prime x of x"), in case of ambiguity of the variable implied by the derivation. Lagrange's notation is sometimes incorrectly attributed to

(f')'=f''\,

and

(f'')'=f'''.

Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df. The second derivative is denoted D²f, and the nth derivative is denoted Dⁿf.

Euler's notation

If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written

D_{x}y\,

or

D_{x}f(x)\,

,

although this subscript is often omitted when the variable x is understood, for instance when this is the only variable present in the expression.

Newton's notation

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If y = f(t), then

{\dot {y}}

and

{\ddot {y}}

denote, respectively, the first and second derivatives of y with respect to t. This notation is used exclusively for derivatives with respect to time or arc length. It is very common in physics, differential equations, and differential geometry.^[1]^[2] While the notation becomes unmanageable for high-order derivatives, in practice only few derivatives are needed.

Elementary rules of differentiation

Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined^[3]^[4]—including complex numbers (C).^[5]

Differentiation is linear

For any functions $f$ and $g$ and any real numbers $a$ and $b$ the derivative of the function $h(x)=af(x)+bg(x)$ with respect to $x$ is

h'(x)=af'(x)+bg'(x).\,

In Leibniz's notation this is written as:

{\frac {d(af+bg)}{dx}}=a{\frac {df}{dx}}+b{\frac {dg}{dx}}.

Special cases include:

The constant factor rule

(af)'=af'\,

The sum rule

(f+g)'=f'+g'\,

The subtraction rule

(f-g)'=f'-g'.\,

The product rule

For the functions f and g, the derivative of the function h(x) = f(x) g(x) with respect to x is

h'(x)=(f(x)g(x))'=f'(x)g(x)+f(x)g'(x).\,

In Leibniz's notation this is written

{\frac {d(fg)}{dx}}={\frac {df}{dx}}g+f{\frac {dg}{dx}}.

The chain rule

The derivative of the function $h(x)=f(g(x))$ with respect to $x$ is

h'(x)=(f(g(x)))'=f'(g(x))\cdot g'(x).\,

In Leibniz's notation this is correctly written as:

{\frac {d}{dx}}h(x)={\frac {d}{dz}}f(z)|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),\,

often abridged to $\quad {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.\,$ Focusing on the notion of maps, and the differential being a map ${\text{D}}$ , this is written in a more concise way as:

[{\text{D}}(h\circ g)]_{x}=[{\text{D}}h]_{g(x)}\cdot [{\text{D}}g]_{x}\,.\,

The inverse function rule

If the function f has an inverse function g, meaning that g(f(x)) = x and f(g(y)) = y, then

g'={\frac {1}{f'\circ g}}.

In Leibniz notation, this is written as

{\frac {dx}{dy}}={\frac {1}{\frac {dy}{dx}}}.

Power laws, polynomials, quotients, and reciprocals

The polynomial or elementary power rule

If $f(x)=x^{r}$ , for any real number $r\neq 0$ then

f'(x)=rx^{r-1}.\,

Special cases include:

If f(x) = x, then f′(x) = 1. This special case may be generalized to:
The derivative of an affine function is constant: if f(x) = ax + b, then f′(x) = a.

Combining this rule with the linearity of the derivative and the addition rule permits the computation of the derivative of any polynomial.

The reciprocal rule

The derivative of h(x) = 1/f(x) for any (nonvanishing) function f is:

h'(x)=-{\frac {f'(x)}{(f(x))^{2}}}.\

In Leibniz's notation, this is written

{\frac {d(1/f)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.\,

The reciprocal rule can be derived from the quotient rule.

The quotient rule

If f and g are functions, then:

\left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}}\quad

wherever g is nonzero.

This can be derived from product rule.

Generalized power rule

The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g,

(f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln f\right),\quad

wherever both sides are well defined.

Special cases:

If f(x) = x^a, f′(x) = ax^{a − 1} when a is any non-zero real number and x is positive.
The reciprocal rule may be derived as the special case where g(x) = −1.

Derivatives of exponential and logarithmic functions

{\frac {d}{dx}}\left(c^{ax}\right)={c^{ax}\ln c\cdot a},\qquad c>0

note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.

{\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}

{\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1

the equation above is also true for all c but yields a complex number if c<0.

{\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.

{\frac {d}{dx}}\left(\ln |x|\right)={1 \over x}.

{\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).

{\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0,{\text{ and if }}{\frac {df}{dx}}{\text{ and }}{\frac {dg}{dx}}{\text{ exist.}}

{\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ if }}f_{i<n}(x)>0{\text{ and }}

{\frac {df_{i}}{dx}}{\text{ exists. }}

Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):

(\ln f)'={\frac {f'}{f}}\quad

wherever f is positive.

Derivatives of trigonometric functions

$(\sin x)'=\cos x\,$	$(\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,$
$(\cos x)'=-\sin x\,$	$(\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}\,$
$(\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,$	$(\arctan x)'={1 \over 1+x^{2}}\,$
$(\sec x)'=\sec x\tan x\,$	$(\operatorname {arcsec} x)'={1 \over \|x\|{\sqrt {x^{2}-1}}}\,$
$(\csc x)'=-\csc x\cot x\,$	$(\operatorname {arccsc} x)'=-{1 \over \|x\|{\sqrt {x^{2}-1}}}\,$
$(\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,$	$(\operatorname {arccot} x)'=-{1 \over 1+x^{2}}\,$

It is common to additionally define an inverse tangent function with two arguments, $\arctan(y,x)$ . Its value lies in the range $[-\pi ,\pi ]$ and reflects the quadrant of the point $(x,y)$ . For the first and fourth quadrant (i.e. $x>0$ ) one has $\arctan(y,x>0)=\arctan(y/x)$ . Its partial derivatives are

{\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}

, and

{\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{2}}}.

Derivatives of hyperbolic functions

$(\sinh x)'=\cosh x={\frac {e^{x}+e^{-x}}{2}}$	$(\operatorname {arsinh} \,x)'={1 \over {\sqrt {x^{2}+1}}}$
$(\cosh x)'=\sinh x={\frac {e^{x}-e^{-x}}{2}}$	$(\operatorname {arcosh} \,x)'={\frac {1}{\sqrt {x^{2}-1}}}$
$(\tanh x)'={\operatorname {sech} ^{2}\,x}$	$(\operatorname {artanh} \,x)'={1 \over 1-x^{2}}$
$(\operatorname {sech} \,x)'=-\tanh x\,\operatorname {sech} \,x$	$(\operatorname {arsech} \,x)'=-{1 \over x{\sqrt {1-x^{2}}}}$
$(\operatorname {csch} \,x)'=-\,\operatorname {coth} \,x\,\operatorname {csch} \,x$	$(\operatorname {arcsch} \,x)'=-{1 \over \|x\|{\sqrt {1+x^{2}}}}$
$(\operatorname {coth} \,x)'=-\,\operatorname {csch} ^{2}\,x$	$(\operatorname {arcoth} \,x)'={1 \over 1-x^{2}}$

Derivatives of special functions

Gamma function $\quad \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,dt$

\Gamma '(x)=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt

\,=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)

\,=\Gamma (x)\psi (x)

with $\psi (x)$ being the digamma function, expressed by the parenthesized expression to the right of $\Gamma (x)$ in the line above.

Riemann Zeta function $\quad \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}$: $\zeta '(x)=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \!$

\,=-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\!

Derivatives of integrals

Suppose that it is required to differentiate with respect to x the function

F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,

where the functions $f(x,t)\,$ and ${\frac {\partial }{\partial x}}\,f(x,t)\,$ are both continuous in both $t\,$ and $x\,$ in some region of the $(t,x)\,$ plane, including $a(x)\leq t\leq b(x),$ $x_{0}\leq x\leq x_{1}\,$ , and the functions $a(x)\,$ and $b(x)\,$ are both continuous and both have continuous derivatives for $x_{0}\leq x\leq x_{1}\,$ . Then for $\,x_{0}\leq x\leq x_{1}\,\,$ :

F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}\,f(x,t)\;dt\,.

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.

Derivatives to nth order

Some rules exist for computing the nth derivative of functions, where n is a positive integer. These include:

Faà di Bruno's formula

If f and g are n times differentiable, then

{\frac {d^{n}}{dx^{n}}}[f(g(x))]=n!\sum _{\{k_{m}\}}^{}f^{(r)}(g(x))\prod _{m=1}^{n}{\frac {1}{k_{m}!}}\left(g^{(m)}(x)\right)^{k_{m}}

where $r=\sum _{m=1}^{n-1}k_{m}$ and the set $\{k_{m}\}$ consists of all non-negative integer solutions of the Diophantine equation $\sum _{m=1}^{n}mk_{m}=n$ .

General Leibniz rule