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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.
If f is a smooth function (a 0-form), then the exterior derivative of f is the differential of f . That is, df is the unique 1-form such that for every smooth vector field X, df (X) = d X f , where d X f is the directional derivative of f in the direction of X.
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
The derivative of the function given by () = + + is ′ = + () () + = + (). Here the second term was computed using the chain rule and the third term using the product rule. The known derivatives of the elementary functions , , (), (), and =, as well as the constant , were also used.
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
if it is zero, then x could be a local minimum, a local maximum, or neither. (For example, f(x) = x 3 has a critical point at x = 0, but it has neither a maximum nor a minimum there, whereas f(x) = ± x 4 has a critical point at x = 0 and a minimum and a maximum, respectively, there.) This is called the second derivative test.
The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule. For example, from the identity A ⋅( B × C ) = ( A × B )⋅ C we may derive A ⋅(∇× C ) = ( A ×∇)⋅ C but not ∇⋅( B × C ) = (∇× B ...
chain rule The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f ∘ g (the function which maps x to f(g(x)) ) in terms of the derivatives of f and g and the product of functions as follows: