Differentiation rules
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This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
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[edit] 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.
[edit] Differentiation is linear
For any functions f and g and any real numbers a and b.
In other words, the derivative of the function h(x) = a f(x) + b g(x) with respect to x is
- h'(x) = a f '(x) + b g '(x).
In Leibniz's notation this is written
Special cases include:
- The constant multiple rule
- The addition rule
- The subtraction rule
[edit] The product or Liebniz rule
For any functions f and g,
In other words, 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).
In Leibniz's notation this is written
[edit] The chain rule
This is a rule for computing the derivative of a function of a function, i.e., of the composite 
of two functions f and g:
In other words, the derivative of the function h(x) = f(g(x)) with respect to x is
- h'(x) = f '(g(x)) g '(x).
In Leibniz's notation this is written (suggestively) as:
[edit] The polynomial or elementary power rule
If f(x) = xn, for some natural number n (including zero) then
Special cases include:
- Constant rule: if f is the constant function f(x) = c, for any real number c, then
- 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
Combining this rule with the linearity of the derivative permits the computation of the derivative of any polynomial.
[edit] The reciprocal rule
For any (nonvanishing) function f, the derivative of the function 1/f (equal at x to 1/f(x)) is
In other words, the derivative of h(x) = 1/f(x) is
- h'(x) = -f '(x)/f(x)2.
In Leibnitz's notation, this is written
[edit] 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
In other words, if y = f(x) has an inverse x = g(y), then
- g'(y) = 1/f '(f-1(y)).
In Leibniz notation, this is written (suggestively) as
[edit] Further rules of differentiation
[edit] The quotient rule
If f and g are functions, then:
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.
[edit] Generalized power rules
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,
wherever both sides are well defined.
[edit] Logarithmic derivatives
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
wherever f is positive.
