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Differential Calculus is to take the derivative of the function. It is important as it can be applied and supports other branches of science. For ex, If you have a velocity function, you can get its acceleration function by taking its derivative, same relationship as well with area and volume formulas.
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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.
ordinary differential equation is obtained only one independent variable and partial differential equation is obtained more than one variable.