Calculus... one of the greatest inventions of mathematics. It is the study of change and is a fundamental concept that has been 'integrated' into the natural world. :P
From projectile motion, to economy, to population gro...
To differentiate polynomial functions we need to learn how to differentiate powers of x. Whether the polynomial is, y = x y = 3x +8 y = 5x³-2x²+9x-15 it can be all easily differentiated using the formula that we will now prove, using First Principles. This formula will also come in handy when dealing with functions where something is to the power of another.
One property of differentiation is that, across operations such as addition, and subtraction, the derivative can be used separately on each term.
In others words, lets say we were differentiating x²+1. Since the terms x² and 1 are separated by addition, the derivative of x² plus the derivative of 1 will be equal to the derivative of x²+1 altogether. This can be shown by:
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Remember, this only applies to addition and subtraction. This does not apply to multiplication and division. The methods for the differentiation of multiplication and division will be explored later.
The proof below requires knowledge of the binomial expansions. It is only used to prove this formula for differentiating powers of x and is not important in later stages of this guide.
To differentiate different powers of x, we need to first devise a general form for indices. Let's say this "power" of x is n. Therefore we have:
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Putting this in an equation gives us:
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