Differentiation from first principles
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Differentiation from First Principles Prerequisites
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In this chapter it is assumed that you are familiar with all the standard derivatives of polynomial, trigonometric, exponential and logarithmic functions; likewise, it is assumed that you are familiar with all the basic rules of differentiation - for example, the Leibniz and chain rules for the differentiation of products and composites of function. You should be familiar with stationary points and how to test for them, both by means of the second derivative and by looking at how the sign of the derivative changes around the stationary point. When you were first introduced to the differential calculus you may have learnt that it arises from the idea of trying to find gradients of functions, or tangents of their graphs. If that ever happened, it may have been now in the misty past, and the purpose of this chapter is to once again make you aware of the logical foundation of the calculus. We are going to call this differentiation from first principles, though as we shall remark at the end, the theory that is presented here itself only represents a stage on the way towards defining a real foundation for the calculus.
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Contents of
Differentiation from first principles
1 Differentiation from First Principles Prerequisites
2 Tangent, Gradient and Rate of Change
3 Differential Calculus
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