We prove our main results by using the time scales chain rule
, the time scales integration by parts formula, the time scales Taylor formula, and classical as well as time scales versions of Holder's inequality.
AD provides a set of mathematically well-defined differentiation rules (symbolic partial derivatives of built-in functions combined with the chain rule
of differential calculus) for transforming a given simulation program into computer code for computing sensitivities.
By recursive application of the chain rule
, the error term for each layer is first calculated.
for Fractional Calculus and Fractional Complex Transform
290, Theorem 2] discovered a fundamental relation from which he deduced the generalized chain rule
for the fractional derivatives.
and using the chain rule
and the fundamental theorem of calculus, we obtain
Using chain rule
: Chain rule
gives the derivative of the composition
Students have experienced difficulty in understanding and using the chain rule
. This study aims at assisting the students to understand and apply the chain rule
and thus inform the author's teaching for future learning of students.
The gradient vector of parameters respect to error function can be obtained by chain rule
in which the optimum values of parameters are determined by iterative algorithms.
For any two [C.sup.1]-functions f and g for which f o g is defined, it is easy to see that the following chain rule
holds, where z = g([zeta]):
The derivative of a composite function, taken with the chain rule
is one of the important notions in calculus.