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.