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}}

Further Reading on Notation

Notation for Differentiation

Further Reading for Rules

Differentiation Rules

Rules of Calculus

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


%d bloggers like this: