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Derivation of x2 or 2x is 2.
The derivation of the formula of pyramid can be gained easily based on the formula for a triangular prism. A pyramid is like two prisms joined together.
applied in making of aeroplane wings
Integration results in an equation which gives the area under the original equation between the bounds. Derivation results in an equation which gives the slope of the original line at any point.
See wikipedia article on polytropic processes.