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This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined^{[1]}^{[2]}—including complex numbers (C).^{[3]}
For any functions f and g and any real numbers a and b the derivative of the function h(x) = af(x) + bg(x) with respect to x is
In Leibniz's notation this is written as:
Special cases include:
For the functions f and g, the derivative of the function h(x) = f(x) g(x) with respect to x is
In Leibniz's notation this is written
The derivative of the function of a function h(x) = f(g(x)) with respect to x is
However, by relaxing the interpretation of h as a function, this is often simply written
If the function f has an inverse function g, meaning that g(f(x)) = x and f(g(y)) = y, then
In Leibniz notation, this is written as
If f(x) = x^n, for any integer n then
Combining this rule with the linearity of the derivative and the addition rule permits the computation of the derivative of any polynomial.
The derivative of h(x) = 1/f(x) for any (nonvanishing) function f is:
In Leibniz's notation, this is written
The reciprocal rule can be derived from the chain rule and the power rule.
If f and g are functions, then:
This can be derived from reciprocal rule and the product rule. Conversely (using the constant rule) the reciprocal rule may be derived from the special case f(x) = 1.
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g,
wherever both sides are well defined.
Special cases:
note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.
the equation above is also true for all c but yields a complex number if c<0.
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
\Gamma'(x) = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt
\zeta'(x) = -\sum_{n=1}^\infty \frac{\ln n}{n^x} = -\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots \!
Suppose that it is required to differentiate with respect to x the function
where the functions f(x,t)\, and \frac{\partial}{\partial x}\,f(x,t)\, are both continuous in both t\, and x\, in some region of the (t,x)\, plane, including a(x)\leq t\leq b(x), x_0\leq x\leq x_1\,, and the functions a(x)\, and b(x)\, are both continuous and both have continuous derivatives for x_0\leq x\leq x_1\,. Then for \,x_0\leq x\leq x_1\,\,:
This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.
Some rules exist for computing the nth derivative of functions, where n is a positive integer. These include:
If f and g are n times differentiable, then
where r = \sum_{m=1}^{n-1} k_m and the set \{k_m\} consists of all non-negative integer solutions of the Diophantine equation \sum_{m=1}^{n} m k_m = n.
These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics. Those in this article (in addition to the above references) can be found in:
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