Calculating Derivatives of General Functions

The direct approach to calculating a derivative (explicitly using Equation 2.1) gets quite tedious for more complicated functions, but fortunately it is virtually never necessary. For example, the functions we encountered in the last chapter have quite simple derivatives  

Incidentally, Equations 2.8 and 2.9 are much simpler than the corresponding equations in other bases  

The extra factor of In 10 = 2.3023. .. in these equations makes base-10 much less convenient. The very simple relationship between an exponential with base e and its derivative is the reason that base is so important, even though e % 2.7183. .. is an irrational number. 

Four general relations are widely used to calculate derivatives of more complicated functions. In the next few equations, /(x) andg(x) are two possibly different functions of x, and C is any numerical constant. All of these relations are discussed in the first semester of calculus. 

The second example above illustrates an important point adding a constant to any function does not change its derivative. 

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