## Derivation Summary

Here is a summary of rules of derivation (sometimes referred to as differentiation).

I wrote this post as the notation used on other sites is often ambiguous (particularly with juxtapositions that could denote either a product or a function application).

I previously wrote some brief notes on some derivations of the derivation formulae.

### Notation in the Literature

Given an expression or function y, there are various notations for the derivative with respect to a variable x.

Newton’s notations included

$\dot{y}$ — in fact this was for derivation with respect to time, t.

Lagrange used

$y^{\prime}$

$y^{(1)}$ — though this form was really only used for higher derivatives

Leibniz used the much more suggestive notation

$\dfrac{\mathrm{d}y}{\mathrm{d}x}$

which had very intuitive manipulations such as

$\dfrac{\mathrm{d}x}{\mathrm{d}y} = 1 / \dfrac{\mathrm{d}y}{\mathrm{d}x}$;

$\dfrac{\mathrm{d}y}{\mathrm{d}u} \, \dfrac{\mathrm{d}u}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}x}$

There is also the operator notation

$\mathrm{D} \, y$

$\mathrm{D}_x \, y$

and the notations for partial derivation (derivation with respect to one of many variables)

$y_x = \dfrac{\partial y}{\partial x}$

### Notation

I will use the notation for partial derivation, and write

$h_x$ for $\dfrac{\mathrm{d} h}{\mathrm{d} x}$

To distinguish between juxtapositions and easily-confused notations, and to establish my conventions, I write

$f \cdot g$ for the product function where $(f \cdot g)(t) = f(t) \times g(t)$

$f \circ g$ for the composite function where $(f \circ g)(t) = f(g(t))$
(note that some authors prefer a reversal of this notation)

There is an ambiguity, or rather, inconsistency, with power notation for functions.

$f^2 = f \cdot f$

is a multiplicative product, where

$f^2(x) = (f(x))^2$

whereas

$f^{-1} = f^{-|}$

is a compositional inverse, not a multiplicative reciprocal, where

$f(f^{-1}(x)) = x$

However, these are so common as conventions, that I do follow them, though preferring the notation $f^{-|}$ for inverse.

### Table

This table summarises some of the rules

$\varphi$
or
$\varphi(x)$
${\varphi}_x$
or
${\varphi}_x(x)$
constant c 0
$x$ 1
$x^n$ $n x^{n-1}$
$f+g$ $f_x+g_x$
$-h$ $-(h_x)$
$f-g$ $f_x-g_x$
$f \cdot g$ $f_x \cdot g + f \cdot g_x$
$\dfrac{1}{h}$ $-\dfrac{h_x}{h^2} = -\dfrac{h_x}{h \cdot h}$
$\dfrac{f}{g}$ $\dfrac{f_x\cdot g-f\cdot g_x}{g^2}$
$f \circ g$ $(f_x \circ g) \cdot g_x$
$h^{-|}$ $\dfrac{1}{h_x \circ h^{-|}}$
$\mathrm{e}^h$ or $\exp \circ h$ $(\mathrm{e}^h) \cdot h_x$
$f^g$ $(f^g) \cdot (\dfrac{f_x \cdot g}{f} + g_x \cdot \ln f)$
$h^n$ $n \, h^{n-1} \cdot h_x$
$c^x$ $c^x \cdot \ln c$
$\ln(h)$ or $\ln \circ h$ $\dfrac{h_x}{h}$

Here are the elementary functions:

$\varphi$
or
$\varphi(x)$
${\varphi}_x$
or
${\varphi}_x(x)$
exp or $\mathrm{e}^x$ exp or $\mathrm{e}^x$ (again)
ln or $\ln x$ $(\lambda x \bullet 1/x)$ or $1/x$
sin cos
cos − sin
tan sec2
cot − csc2
sec $\sec \cdot \tan$
csc $- \csc \cdot \cot$
$\arcsin x$ $\dfrac{1}{\sqrt{1-x^2}}$
$\arccos x$ $- \dfrac{1}{\sqrt{1-x^2}}$
$\arctan x$ $\dfrac{1}{1+x^2}$
$\mathrm{arccot} x$ $- \dfrac{1}{1+x^2}$
$\mathrm{arcsec} x$ $\dfrac{1}{|x| \cdot \sqrt{x^2-1}}$
$\mathrm{arccsc} x$ $- \dfrac{1}{|x| \cdot \sqrt{x^2-1}}$
sinh cosh
cosh sinh
tanh sech2
coth − csch2
sech $- \mathrm{sech} \cdot \tanh$
csch $- \mathrm{csch} \cdot \mathrm{coth}$
$\mathrm{arsinh} x$ $\dfrac{1}{\sqrt{x^2+1}}$
$\mathrm{arcosh} x$ $\dfrac{1}{\sqrt{x^2-1}}$
$\mathrm{artanh} x$ $\dfrac{1}{1-x^2}$
$\mathrm{arcoth} x$ $-\dfrac{1}{1-x^2}$
$\mathrm{arsech} x$ $- \dfrac{1}{x \sqrt{1-x^2}}$
$\mathrm{arcsch} x$ $- \dfrac{1}{|x| \sqrt{1+x^2}}$