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In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and () The quotient rule states that the derivative of h(x) is
This is the definition of the derivative. All differentiation rules can also be reframed as rules involving limits. For example, if g(x) is differentiable at x, (+) = ′ [()] ′ (). This is the chain rule.
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
Two other well-known examples are when integration by parts is applied to a function expressed as a product of 1 and itself. This works if the derivative of the function is known, and the integral of this derivative times is also known. The first example is (). We write this as:
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 ...
Finding the derivative of an expression is a straightforward process for which it is easy to construct an algorithm. The reverse question of finding the integral is much more difficult. Many expressions that are relatively simple do not have integrals that can be expressed in closed form.
Let = be an infinite series with real terms and let : be any real function such that (/) = for all positive integers n and the second derivative ″ exists at =. Then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges absolutely if f ( 0 ) = f ′ ( 0 ) = 0 {\displaystyle f(0)=f'(0)=0} and diverges otherwise.
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.)