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In calculus, the differential represents the principal part of the change in a function = with respect to changes in the independent variable. The differential is defined by = ′ (), where ′ is the derivative of f with respect to , and is an additional real variable (so that is a function of and ).
If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at ( x , y ) implies that f xx and f yy have the same sign there.
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 ...
Equivalently, the slope could be estimated by employing positions x − h and x. Another two-point formula is to compute the slope of a nearby secant line through the points (x − h, f(x − h)) and (x + h, f(x + h)). The slope of this line is (+) ().
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 ()
At each point p, this linear map from R n to R is denoted df p and called the derivative or differential of f at p. Thus df p ( v ) = ∂ v f ( p ) . Extended over the whole set, the object df can be viewed as a function that takes a vector field on U , and returns a real-valued function whose value at each point is the derivative along the ...
The Poincaré lemma states that if B is an open ball in R n, any closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n. [ 1 ] More generally, the lemma states that on a contractible open subset of a manifold (e.g., R n {\displaystyle \mathbb {R} ^{n}} ), a closed p -form, p > 0, is exact.
Variable changes for differentiation and integration are taught in elementary calculus and the steps are rarely carried out in full. The very broad use of variable changes is apparent when considering differential equations, where the independent variables may be changed using the chain rule or the dependent variables are changed resulting in ...