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  2. 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 f {\displaystyle f} and g {\displaystyle g} be n {\displaystyle n} -times differentiable functions. The base case when n = 1 {\displaystyle n=1} claims that: ( f g ) ′ = f ′ g + f g ′ , {\displaystyle (fg)'=f'g+fg',} which is the usual product rule and is known ...

  3. Notation for differentiation - Wikipedia

    en.wikipedia.org/wiki/Notation_for_differentiation

    If f is a function, then its derivative evaluated at x is written ′ (). It first appeared in print in 1749. [3] Higher derivatives are indicated using additional prime marks, as in ″ for the second derivative and ‴ for the third derivative. The use of repeated prime marks eventually becomes unwieldy.

  4. 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. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and its angle is zero. In other words, the value of the constant function, y, will not change as the value of x increases or decreases.

  5. King James Only movement - Wikipedia

    en.wikipedia.org/wiki/King_James_Only_movement

    The exlusive use of the King James Version is recorded in a statement made by the Tennessee Association of Baptists in 1817, stating "We believe that any person, either in a public or private capacity who would adhere to, or propagate any alteration of the New Testament contrary to that already translated by order of King James the 1st, that is now in common in use, ought not to be encouraged ...

  6. Finite difference coefficient - Wikipedia

    en.wikipedia.org/wiki/Finite_difference_coefficient

    For arbitrary stencil points and any derivative of order < up to one less than the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations [6] ( s 1 0 ⋯ s N 0 ⋮ ⋱ ⋮ s 1 N − 1 ⋯ s N N − 1 ) ( a 1 ⋮ a N ) = d !

  7. Inverse function rule - Wikipedia

    en.wikipedia.org/wiki/Inverse_function_rule

    In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...

  8. Differential calculus - Wikipedia

    en.wikipedia.org/wiki/Differential_calculus

    Still better might be a cubic polynomial a + b(xx 0) + c(xx 0) 2 + d(xx 0) 3, and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients a , b , c , and d that makes the approximation as good as possible.

  9. Arctangent series - Wikipedia

    en.wikipedia.org/wiki/Arctangent_series

    The derivative of arctan x is 1 / (1 + x 2); conversely, the integral of 1 / (1 + x 2) is arctan x.. If = ⁡ then ⁡ =. The derivative is = ⁡ = + ⁡. Taking the reciprocal, = + ⁡ = +.