Differentiation - used to find the rate of change of a function (the gradient). For a function f(x) = axⁿ, f'(x) = naxⁿ⁻¹. For example, the function y = 3x²+ 24x + 10 differentiates into dy/dx = 6x + 24. The rule is: Multiply the coefficient by the exponent, then decrease the exponent by 1.
Integration - used to find the area under a curve. By integrating a differentiated function, you return to the original function. Therefore, the method of integration is quite logically following. Increase the exponent by 1, then divide by the new exponent.
∫f'(x) dx = f(x) + C
where C is the constant of integration. It cannot be determined unless we know one coordinate through which f(x) passes.
This is called the fundamental theorem of calculus
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