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  2. Mean value theorem (divided differences) - Wikipedia

    en.wikipedia.org/wiki/Mean_value_theorem...

    For any n + 1 pairwise distinct points x ... , x n in the domain of an n-times differentiable ... where the nth derivative of f equals n ! times the nth divided ...

  3. Differential (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Differential_(mathematics)

    For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is, intuitively, extremely useful, and there are a number of ways to make the notion mathematically ...

  4. Differentiable function - Wikipedia

    en.wikipedia.org/wiki/Differentiable_function

    It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable. If f is differentiable at a point x 0, then f must also be continuous at x 0. In particular, any differentiable function must ...

  5. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    The absolute value function is continuous but fails to be differentiable at x = 0 since the tangent slopes do not approach the same value from the left as they do from the right. If f {\displaystyle f} is differentiable at ⁠ a {\displaystyle a} ⁠ , then f {\displaystyle f} must also be continuous at a {\displaystyle a} . [ 11 ]

  6. Differential of a function - Wikipedia

    en.wikipedia.org/wiki/Differential_of_a_function

    the partial differential of y with respect to any one of the variables x 1 is the principal part of the change in y resulting from a change dx 1 in that one variable. The partial differential is therefore involving the partial derivative of y with respect to x 1.

  7. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.

  8. Total derivative - Wikipedia

    en.wikipedia.org/wiki/Total_derivative

    The rate of change of f with respect to x is usually the partial derivative of f with respect to x; in this case, ∂ f ∂ x = y . {\displaystyle {\frac {\partial f}{\partial x}}=y.} However, if y depends on x , the partial derivative does not give the true rate of change of f as x changes because the partial derivative assumes that y is fixed.

  9. Smoothness - Wikipedia

    en.wikipedia.org/wiki/Smoothness

    The C 0 function f (x) = x for x ≥ 0 and 0 otherwise. The function g (x) = x 2 sin(1/ x) for x > 0. The function : with () = ⁡ for and () = is differentiable. However, this function is not continuously differentiable.

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