Differentiation

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

Main article: Linearity of differentiation

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

In Leibniz's notation this is written

Special cases include:

* The constant multiple rule

* The sum rule

* The subtraction rule

The product or Leibniz rule

Main article: Product rule

For any of the functions f and g,

In other words, the derivative of the function h(x) = f(x) * g(x) with respect to x is

In Leibniz's notation this is written

The chain rule

Main article: 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

In Leibniz's notation this is written (suggestively) as:

The polynomial or elementary power rule

Main article: Calculus with polynomials

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 number c, then for all x

* 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.

The reciprocal rule

Main article: 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

In Leibniz's notation, this is written

The inverse function rule

Main article: inverse functions and differentiation

This should not be confused with the...