Well generally, when you learn basic calculus such as that is stated above, without the integral sign dx on its own it's just lazy notation and not rigorously defined. It's just a bit of time and hides a bunch of results in it such as the Fundamental Theorems of Calculus:G01 said:yeah but i don't need to know how they work but why you can treat the dx in an integral as a differential. I know it sounds weird bear with me.
Differentials in integrals refer to the infinitesimal changes in the independent variable (usually denoted by dx) that are used to calculate the change in the dependent variable (usually denoted by dy) in a given integral. They are an important concept in calculus and are used to find the area under a curve or the volume of a solid.
Differentials are used in integrals to represent small changes in the independent variable, which are then multiplied by the derivative of the function to calculate the change in the dependent variable. This allows for the calculation of areas and volumes that would otherwise be impossible to find using basic geometry.
A differential (dx) is an infinitesimal change in the independent variable, while a derivative (dy/dx) is the ratio of the change in the dependent variable to the change in the independent variable. In other words, a derivative is the rate of change of a function, while a differential is the actual change in the variable.
Differentials are important in calculus because they allow for the calculation of areas and volumes under curves, which is essential in many real-world applications such as physics, engineering, and economics. They also help simplify complex integrals and make them easier to solve.
Yes, differentials can be used to approximate values of functions. This is known as linear approximation, where the derivative of the function at a specific point is used to estimate the value of the function at that point. This method is especially useful when calculating values for functions that are difficult to evaluate directly.