You learn the rules for differentiating polynomials, products, quotients, etc. Then you learn the chain rule and a couple of other rules and you're good to go for the basics. You can check your results by learning to use wolframalpha.com.
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Isaac Newton and/or Gottfried Wilhelm von Leibniz, depending upon whom you ask.
Calculus catch phrases:Dealing with two fundamental operations, differentiation and integration, carried out on functions.Purely theoretical aspects of these operations and their interrelation.Standard functionsDerivative of a function of one variable
In short, no. Elementary calculus includes finding limits, basic differentiation and integration, dealing with sequences and series, and simple vector operations, among other concepts. Pre-calculus mostly focuses on the algebra necessary to perform those operations, with perhaps some introduction to limits or other simple ideas from elementary calculus.
Pretty much any serious statistical model or experiment on anything will use basic calculus to interpret data. Anything that exponentially grows or decays (radioactive matter, bacteria, population growth, etc.) Anything that's built to be structurally sound. Anything that uses the EM spectra (radio, microwaves, visible light, etc.) All scientific industries use calculus practically constantly. And on and on and on... In reality, it's rarely pure theoretical calculus that's being done. Rather, another branch of math based on and built from the principles and results of calculus is primarily used called differential equations. Don't forget integration, the other "half" of calculus. That is as equally important in your listed applications. Also, both theoretical and applied calculus use both differentiation and integration. Differentiation isn't a separate branch of maths, but one of the two major branches of calculus as a whole.
No calculus is harder, because calculus is basically a combination of algebra and trigonometry, so you need algebra to do calculus. Also, calculus involves limits, differentiation, and integration. Integration makes algebra look like kindergarten. +++ Meaningless question, ditto with the answers I'm afraid. These are not separate entities but all fields of mathematics, and you use algebra in expressing and solving mathematical problems. Calculus is NOT "basically a combination of algebra and trigonometry". You can differentiate and integrate trig. functions, but although calculus alone does not rely on trigonometry for its existence, its manoeuvres are all algebraic steps. As to comparative difficulty, that is entirely down to you. If you find algebra difficult you will find trigonometry and calculus difficult, because algebra is used to describe those two (and any other) mathematical process. Algebra is not an isolated topic!