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In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after ...
Calculus. In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. [1][2][3] Let , where both f and g are differentiable and The quotient rule states that the derivative of h(x) is. It is provable in many ways by using other derivative rules.
The Lie derivative is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold). It is a grade 0 derivation on the algebra.
Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field. As such, it does not generalize to other fields, but the following corollary does: if a real polynomial factors (has all of its roots) over the real numbers, then its derivative does as well.
Derivative test. In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. The usefulness of derivatives to find extrema is ...
t. e. 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 chain rule may also be expressed in ...