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And for simple expressions, you just try all the rules.
For simple expressions such as 4x^3, to obtain the derivative, one takes the exponent down and multiplies it as a (combined) coefficient, and reduces the exponent by 1 power, so the derivative of x^n = nx^ n-1) anx^ n-1derivandve of 4x^3 = 12x^2.
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With the power of a computerized formula manipulator which can handle algebraic and trigonometric expressions, the development of simple expressions for coordinates and elements from the existing analytic theories is now feasible.
In fact I suggest that the descriptive conception of Fregean sense is universally wrong, at least for semantically simple expressions.
(see the Appendix for their simple expressions) and r k ( n ) : = ∑ m = 1 n 1 m − ln n − R k ( n ).
Using nonlinear wave propagation theory for distillation processes, we derive simple expressions for the prediction of multiplicity regions in parameter space with paper and pencil.
For specific pressure time histories, simple expressions for the resulting seismicity density can be derived.
For example, we find the following simple expressions for the cluster coefficient and the topological overlap.
Furthermore, simple expressions for mean Nusselt number, valid for the range of parameters embraced in the present study, are also provided.
Simple expressions for the modal density of fluid filled pipes and for the input power from mechanical and fluid point sources are derived.
The obtained results provide simple expressions for tuning these controllers.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com