Differentials can be used to approximate a nonlinear function as a linear function.
They can be used as a "factory" to quickly find partial derivatives.
They can be used to test if a function is smooth.
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ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.
As an automotive engineer, I mostly use algebra, but I sometimes use geometry, statistics, and calculus. Some higher level research positions may use differential equations.
The mathematician who formalised differential calculus and who is responsible for the notation in use today is Gottfried Leibniz. However, in has calculation of pi, Aristotle used the principle of convergent series and limits more than 2000 years earlier.
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Finite Differential Methods (FDM) are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.