Basic Formulas of Differential Calculus

Generally, the definite integral exists for functions that have at most a finite number of finite discontinuities—classified as piecewise continuous. Most often an integral does not exist if it blows up to an infinite value, for example, Jq x dx. These are also known as improper integrals. 

The terms differentiating and finding the derivative are synonymous. A few simple rules suffice to determine the derivatives of most functions you will encounter. These can usually be deduced from the definition of derivative in Eq. (6.12). Consider first the function y(x) = ax , where a is a constant. We will need 

The first formula means that the derivative of a constant is zero. Eq. (6.21) is also valid for fractional or negative values of n. Thus, we find 

the exponential function equals its own derivative. This result also follows from term-hy-term differentiation of the series (3.104). The result 

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