Differentiation Rules (mathematics)
Differentiation rules are fundamental principles in calculus that simplify the process of finding the derivative of functions. Developed in the late seventeenth century by mathematicians like Johann Bernoulli and Gottfried Leibniz, these rules emerged from the need to apply the limit definition of a derivative more effectively. Key rules include the addition rule, which asserts that the derivative of a sum is the sum of the derivatives, and the constant multiple rule, which allows constants to be factored out of derivatives. The power rule is particularly significant, enabling the differentiation of polynomial expressions. Other important rules include the product and quotient rules for differentiating products and ratios of functions, respectively.
Additionally, differentiation of trigonometric functions relies on the derivatives of sine and cosine, leading to a structured approach for finding the derivatives of other trigonometric identities. The chain rule facilitates differentiation of composite functions, allowing for more complex equations to be handled effectively. Furthermore, implicit differentiation is a valuable technique for deriving relationships expressed implicitly, while logarithmic differentiation aids in tackling more complicated functions. Collectively, these rules form a comprehensive toolkit for solving a wide range of differentiation problems in mathematics.
On this Page
Subject Terms
Differentiation Rules (mathematics)
The differentiation rules are a set of rules found in the late seventeenth century by mathematicians such as Johann Bernoulli and Gottfried Leibniz to aid in finding the derivative of a function.
Using limits, we can define the derivative of a function
as follows:
.
However, this formulation is difficult to work with for most functions. As mathematicians worked with derivatives in the late seventeenth century, several differentiation rules were found to simplify this process.
There are two rules that follow immediately from the above definition and properties of limits. The first of these is the addition rule which states that
.
In other words, the derivative of a sum is the sum of the derivatives. The second such rule is the constant multiple rule. This states that if
is a constant and
is a differentiable function, then
A constant can be pulled outside of the derivative operator.
One of the most important rules of differentiation is the power rule. The power rule follows from Cavalieri’s quadrature formula. The power rules states that
.
There are certain functions that need to be rewritten in order to make the power rule more transparent. For example,
and
. Once these functions are rewritten, the power rule can easily be applied.
One of the most important functions in all of mathematics is the natural exponential function,
. Johann Bernoulli discovered the differentiation rule for this function in 1697. Namely, he discovered that
. He also discovered the derivative of its inverse function, the natural logarithm. The differentiation rule associated with the natural logarithm is
.
If two functions are multiplied together, then the derivative of the result can be obtained using the product rule discovered by Gottfried Leibniz. The product rule states that for differentiable functions
and
,
The corresponding quotient rule for the ratio of two differentiable functions follows immediately from the product rule and elementary algebra. The quotient rule states that
The differentiation rules for the trigonometric functions are based on the derivatives of sine and cosine. The corresponding derivatives are
and
, respectively. The derivatives of the remaining trigonometric functions can be obtained using a combination of the rules for sine and cosine, the quotient rule, and the Pythagorean identity
The differentiation rules for the remaining trigonometric functions are as follows:
The next important differentiation rule is the chain rule. The chain rule is usually attributed to Leibniz, however there is evidence that suggests that it is in fact due to Isaac Barrow. The chain rule allows for the differentiation of a composition function
. The chain rule states that
.
Many curves in mathematics are difficult (or even impossible) to express as an explicit function
. Such curves include the unit circle,
, as well more esoteric examples such as
. If we consider
as an implicit function of
, then we can take the derivative of both sides of such equations with respect to
. When doing so, we must use the chain rule to differentiate any functions that involve
. For example,
. Applying this idea to the desired curve results in an algebraic expression that can be solved for
.
Implicit differentiation can be used to find the derivatives of additional functions, such as
(also called
). Observe that if
, then
. At which point, determining the derivative is an implicit differentiation problem. This method allows us to compute the derivatives of the six inverse trigonometric functions:
Logarithmic differentiation is a similar technique that is useful for differentiating functions such as
. If the natural logarithm is applied to both sides of this equation, then we obtain the equation
using properties of logarithms. This reduces the problem to an implicit differentiation problem.
Additional Readings
Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. New York: Wiley, 2010. Print.
Gardner, Robert B. "A Useful Notation for Rules of Differentiation." College Mathematics Journal. Vol. 24, No. 4, p. 351–52, (1993). Print.
Hass, Joel; Maurice Weir, and George B. Thomas. Thomas’ Calculus Early Transcendentals. 13th ed. Boston: Pearson, 2013. Print.