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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.
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.
In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. The chain rule is also valid for Fréchet derivatives in Banach spaces.
Differentiation rules – Rules for computing derivatives of functions General Leibniz rule – Generalization of the product rule in calculus Inverse functions and differentiation – Calculus identity Pages displaying short descriptions of redirect targets
Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as differentiation. [28]
The partial derivative of f with respect to x does not give the true rate of change of f with respect to changing x because changing x necessarily changes y. However, the chain rule for the total derivative takes such dependencies into account. Write () = (, ()). Then, the chain rule says
However, because integration is the inverse operation of differentiation, Lagrange's notation for higher order derivatives extends to integrals as well. Repeated integrals of f may be written as f ( − 1 ) ( x ) {\displaystyle f^{(-1)}(x)} for the first integral (this is easily confused with the inverse function f − 1 ( x ) {\displaystyle f ...
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.