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1.2 Example 2: Derivative of tangent function. 2 Reciprocal ... the quotient rule is a method of finding the derivative of a function that is the ratio of two ...
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified ...
The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable but only says what its derivative is if it is differentiable.)
The chain rule can be used to derive some well-known differentiation rules. For example, the quotient rule is a consequence of the chain rule and the product rule. To see this, write the function f(x)/g(x) as the product f(x) · 1/g(x).
For example, the derivative of the sine function is written sin ′ (a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle. All derivatives of circular trigonometric functions can be found from those of sin( x ) and cos( x ) by means of the quotient rule applied to functions such ...
In summary, both derivatives and logarithms have a product rule, a reciprocal rule, a quotient rule, and a power rule (compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.
Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. [2] The first three operations below assume that x = b c and/or y = b d, so that log b (x) = c and log b (y) = d.