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  2. Product rule - Wikipedia

    en.wikipedia.org/wiki/Product_rule

    In this terminology, the product rule states that the derivative operator is a derivation on functions. In differential geometry , a tangent vector to a manifold M at a point p may be defined abstractly as an operator on real-valued functions which behaves like a directional derivative at p : that is, a linear functional v which is a derivation ...

  3. General Leibniz rule - Wikipedia

    en.wikipedia.org/wiki/General_Leibniz_rule

    The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let and be -times differentiable functions.The base case when = claims that: ′ = ′ + ′, which is the usual product rule and is known to be true.

  4. Derivative - Wikipedia

    en.wikipedia.org/wiki/Derivative

    The derivative of the function given by () = + ⁡ ⁡ + is ′ = + ⁡ (⁡) ⁡ () + = + ⁡ ⁡ (). Here the second term was computed using the chain rule and the third term using the product rule. The known derivatives of the elementary functions , , ⁡ (), ⁡ (), and ⁡ =, as well as the constant , were also used.

  5. Triple product rule - Wikipedia

    en.wikipedia.org/wiki/Triple_product_rule

    Suppose a function f(x, y, z) = 0, where x, y, and z are functions of each other. Write the total differentials of the variables = + = + Substitute dy into dx = [() + ()] + By using the chain rule one can show the coefficient of dx on the right hand side is equal to one, thus the coefficient of dz must be zero () + = Subtracting the second term and multiplying by its inverse gives the triple ...

  6. Derivation (differential algebra) - Wikipedia

    en.wikipedia.org/wiki/Derivation_(differential...

    The partial derivative with respect to a variable is an R-derivation on the algebra of real-valued differentiable functions on R n. The Lie derivative with respect to a vector field is an R-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold

  7. Jacobi identity - Wikipedia

    en.wikipedia.org/wiki/Jacobi_identity

    The simplest informative example of a Lie algebra is constructed from the (associative) ring of matrices, which may be thought of as infinitesimal motions of an n-dimensional vector space. The × operation is the commutator, which measures the failure of commutativity in matrix multiplication.

  8. How Democrats are planning to fight Trump's mass deportation plan

    www.aol.com/immigration-litigation-being...

    Immigration advocacy groups and Democratic leaders are seeking to disrupt President-elect Donald Trump's plan to deport millions of undocumented immigrants by pre-drafting lawsuits that could be ...

  9. Differentiation rules - Wikipedia

    en.wikipedia.org/wiki/Differentiation_rules

    The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): (⁡) ′ = ′ wherever f is positive. ...

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