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  2. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.

  3. Functional derivative - Wikipedia

    en.wikipedia.org/wiki/Functional_derivative

    In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative. For example, consider the functional [] = (, (), ′ ()), where f ′(x) ≡ df/dx.

  4. Chain rule - Wikipedia

    en.wikipedia.org/wiki/Chain_rule

    For example, in the manifold case, the derivative sends a C r-manifold to a C r−1-manifold (its tangent bundle) and a C r-function to its total derivative. There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives.

  5. Total derivative - Wikipedia

    en.wikipedia.org/wiki/Total_derivative

    In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function. [1]: 198–203

  6. Quotient rule - Wikipedia

    en.wikipedia.org/wiki/Quotient_rule

    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 () The quotient rule states that the derivative of h(x) is

  7. Matrix calculus - Wikipedia

    en.wikipedia.org/wiki/Matrix_calculus

    By example, in physics, the electric field is the negative vector gradient of the electric potential. The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows.

  8. 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.

  9. Weak derivative - Wikipedia

    en.wikipedia.org/wiki/Weak_derivative

    The absolute value function : +, = | |, which is not differentiable at = has a weak derivative : known as the sign function, and given by () = {>; =; < This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. For example, the definition of v(0) above could be replaced with any ...