Definitions

Differentiation rules

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

Elementary rules of differentiation

Unless otherwise stated, all functions will be functions from R to R, although more generally, the formulae below make sense wherever they are well defined.

Differentiation is linear

For any functions f and g and any real numbers a and b.

$\left(af + bg\right)\text{'} = af\text{'} + bg\text{'}.,$
In other words, the derivative of the function h(x) = a f(x) + b g(x) with respect to x is

$h\text{'}\left(x\right) = a f\text{'}\left(x\right) + b g\text{'}\left(x\right).,$
In Leibniz's notation this is written
$frac\left\{d\left(af+bg\right)\right\}\left\{dx\right\} = afrac\left\{df\right\}\left\{dx\right\} +bfrac\left\{dg\right\}\left\{dx\right\}.$

Special cases include:

$\left(af\right)\text{'} = a,f\text{'} ,$

$\left(f + g\right)\text{'} = f\text{'} + g\text{'},$

• The subtraction rule

$\left(f - g\right)\text{'} = f\text{'} - g\text{'}.,$

The product or Leibniz rule

For any functions f and g,

$\left(fg\right)\text{'} = f\text{'} g + f g\text{'}.,$
In other words, the derivative of the function h(x) = f(x) g(x) with respect to x is
$h\text{'}\left(x\right) = f\text{'}\left(x\right) g\left(x\right) + f\left(x\right) g\text{'}\left(x\right).,$
In Leibniz's notation this is written
$frac\left\{d\left(fg\right)\right\}\left\{dx\right\} = frac\left\{df\right\}\left\{dx\right\} g + f frac\left\{dg\right\}\left\{dx\right\}.$

The chain rule

This is a rule for computing the derivative of a function of a function, i.e., of the composite $fcirc g$ of two functions f and g:

$\left(f circ g\right)\text{'} = \left(f\text{'} circ g\right)g\text{'}.,$
In other words, the derivative of the function h(x) = f(g(x)) with respect to x is
$h\text{'}\left(x\right) = f\text{'}\left(g\left(x\right)\right) g\text{'}\left(x\right).,$
In Leibniz's notation this is written (suggestively) as:
$frac\left\{df\right\}\left\{dx\right\} = frac\left\{df\right\}\left\{dg\right\} frac\left\{dg\right\}\left\{dx\right\}.,$

The polynomial or elementary power rule

If $f\left(x\right) = x^n$, for some natural number n (including zero) then

$f\text{'}\left(x\right) = nx^\left\{n-1\right\}.,$

Special cases include:

• Constant rule: if f is the constant function f(x) = c, for any number c, then for all x

$f\text{'}\left(x\right) = 0.,$

• The derivative of a linear function is constant: if f(x) = ax (or more generally, in view of the constant rule, if f(x)=ax+b ), then

$f\text{'}\left(x\right) = a.,$

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

The reciprocal rule

For any (nonvanishing) function f, the derivative of the function 1/f (equal at x to 1/f(x)) is

$-frac\left\{f\text{'}\right\}\left\{f^2\right\}.,$
In other words, the derivative of h(x) = 1/f(x) is

$h\text{'}\left(x\right) = -frac\left\{f\text{'}\left(x\right)\right\}\left\{f\left(x\right)^2\right\}.$

In Leibniz's notation, this is written

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

The inverse function rule

This should not be confused with the reciprocal rule: the reciprocal 1/x of a nonzero real number x is its inverse with respect to multiplication, whereas the inverse of a function is its inverse with respect to function composition.

If the function f has an inverse g = f−1 (so that g(f(x)) = x and f(g(y)) = y) then

$g\text{'} = frac\left\{1\right\}\left\{f\text{'}circ f^\left\{-1\right\}\right\}.,$

In Leibniz notation, this is written (suggestively) as

$frac\left\{dx\right\}\left\{dy\right\} = frac\left\{1\right\}\left\{dy/dx\right\}.$

Further rules of differentiation

The quotient rule

If f and g are functions, then:

$left\left(frac\left\{f\right\}\left\{g\right\}right\right)\text{'} = frac\left\{f\text{'}g - fg\text{'}\right\}\left\{g^2\right\}quad$ wherever g is nonzero.

This can be derived from reciprocal rule and the product rule. Conversely (using the constant rule) the reciprocal rule is the special case f(x) = 1.

Generalized power rule

The elementary power rule generalizes considerably. First, if x is positive, it holds when n is any real number. The reciprocal rule is then the special case n = -1 (although care must then be taken to avoid confusion with the inverse rule).

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

$\left(f^g\right)\text{'} = left\left(e^\left\{gln f\right\}right\right)\text{'} = f^gleft\left(f\text{'}\left\{g over f\right\} + g\text{'}ln fright\right),quad$
wherever both sides are well defined.

Logarithmic derivatives

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

$\left(ln f\right)\text{'}= frac\left\{f\text{'}\right\}\left\{f\right\} quad$ wherever f is positive.