Waves/Derivatives

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Math Tutorial -- Derivatives

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Figure 1.15: Estimation of the derivative
Figure 1.15: Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A.

This section provides a quick introduction to the idea of the derivative. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation.

Often we are interested in the slope of a line tangent to a function at some value of . This slope is called the derivative and is denoted . Since a tangent line to the function can be defined at any point , the derivative itself is a function of :

(2.25)

As figure 1.15 illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve:

(2.26)

As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.

Table of Derivatives

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Derivatives of some common functions are now given. In each case   is a constant.


Table of Derivatives
 
 
 
 
  where both xc and cxc−1 are defined.
 
 
  x > 0
  c > 0
 
  c > 0, c ≠ 1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

The product and chain rules are used to compute the derivatives of complex functions. For instance,

 

and

 

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