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A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usual difference quotient. [3]
A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists.
For example, the function is symmetrically continuous at =, but not continuous. Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability.
The symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied (the second partial derivatives are not continuous). The function f(x, y), as shown in equation , does not have symmetric second derivatives at its origin.
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 ]
The character ∂ (Unicode: U+2202) is a stylized cursive d mainly used as a mathematical symbol, usually to denote a partial derivative such as / (read as "the partial derivative of z with respect to x").
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If f is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points. If f is twice differentiable, then conversely, a critical point x of f can be analysed by considering the second derivative of f at x : if it is positive, x is a local minimum; if it is negative, x is a ...