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Preface

My first experience with derivatives was seeing how they are obtained from the usual definition

$$f'(x)=\underset{\text{$\Delta $x}\to 0}{\text{Lim}}\frac{f (\text{$x+\Delta $x})-f (x)}{\text{$\Delta $x}}.$$

I accepted the binomial theorem derivation in the case of polynomials and the small-angle explanation in the case of sines and cosines until my math instructor asserted, without justification, that the derivative of the exponential is itself. I had to wait until Taylor expansions in order to understand the exponential derivative. At that time I also saw the series for the cosine and sine, which up to that point I knew as the ratio of one or the other right side to the hypotenuse in a right triangle. I reached graduate school having stored the trigonometric functions, the unit circle, the right triangle side ratios, and the series expansions in appropriate compartments in my brain, ready to use them correctly but not really bothering to understand how and if they were...

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