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A differentiable function. In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
Product rule: For two differentiable functions f and g, () = +. An operation d with these two properties is known in abstract algebra as a derivation . They imply the power rule d ( f n ) = n f n − 1 d f {\displaystyle d(f^{n})=nf^{n-1}df} In addition, various forms of the chain rule hold, in increasing level of generality: [ 12 ]
If a continuous function on an open interval (,) satisfies the equality () =for all compactly supported smooth functions on (,), then is identically zero. [1] [2]Here "smooth" may be interpreted as "infinitely differentiable", [1] but often is interpreted as "twice continuously differentiable" or "continuously differentiable" or even just "continuous", [2] since these weaker statements may be ...
Most functions that occur in practice have derivatives at all points or almost every point. Early in the history of calculus, many mathematicians assumed that a continuous function was differentiable at most points. [14] Under mild conditions (for example, if the function is a monotone or a Lipschitz function), this is true. However, in 1872 ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
Let be a function in the Lebesgue space ([,]).We say that in ([,]) is a weak derivative of if ′ = ()for all infinitely differentiable functions with () = =.. Generalizing to dimensions, if and are in the space () of locally integrable functions for some open set, and if is a multi-index, we say that is the -weak derivative of if
Here u(x,t) is the temperature of the rod at position x and time t and ... The mean value theorem: For each differentiable function : [,] ...
The reason why there is no analog of mean value equality is the following: If f : U → R m is a differentiable function (where U ⊂ R n is open) and if x + th, x, h ∈ R n, t ∈ [0, 1] is the line segment in question (lying inside U), then one can apply the above parametrization procedure to each of the component functions f i (i = 1 ...