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This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ...
Calculus [from Latin, literally pebble (used in reckoning)] is a major area in mathematics which relates small-scale phenomena with large-scale behavior. ...
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. In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...
Differentiation is linear -
For any functions f and g and any real numbers a and b. In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. ...
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 In calculus, Leibnizs notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz, was originally the use of expressions such as dx and dy and to represent infinitely small (or infinitesimal) increments of quantities x and y, just as Δx and Δy represent finite...
Special cases include: In calculus, the constant factor rule in differentiation allows you to take constants outside a derivative and concentrate on differentiating the function of x itself. ...
The sum rule in differentiation is possibly the most useful rule in differentiation. ...
The product or Liebniz rule -
For any functions f and g, In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...
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 -
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 calculus, the chain rule is a formula for the derivative of the composite of two functions. ...
The term composite can refer to several different things: A dental composite is an type of tooth filling material made of a plastic matrix containing high-strength quartz filler particles. ...
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 -
If f(x) = xn, for some natural number n (including zero) then In mathematics, polynomials are perhaps the simplest functions with which to do calculus. ...
In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
Special cases include: - Constant rule: if f is the constant function f(x) = c, for any real 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 -
For any (nonvanishing) function f, the derivative of the function 1/f (equal at x to 1/f(x)) is In calculus, the reciprocal rule is a shorthand method of finding the derivative of a function that is the reciprocal of a differentiable function, without using the quotient rule or chain rule. ...
In other words, the derivative of h(x) = 1/f(x) is In Leibnitz's notation, this is written
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. In mathematics, the inverse of a function is a function that, in some fashion, undoes the effect of (see inverse function for a formal and detailed definition). ...
In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
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 In Leibniz notation, this is written (suggestively) as
Further rules of differentiation
The quotient rule -
If f and g are functions, then: In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ...
- 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 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). In mathematics, the power rule is a method for differentiating expressions involving exponentiation (the power operation). ...
The most general power rule is the functional power rule: for any functions f and g,
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): In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula f′/f where f′ is the derivative of f. ...
- wherever f is positive.
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