Derivatives

Increments:  The increment of a variable x is the change in x as it increases or decreases from one value x₀ to another x₁ in its range. The change in value is represented by Dx and so:  Dx = x₁ – x₀ or, rearranging:  x₁ = x₀ + Dx

If x is given an increment then a function y = f (x) will also be given an increment, which we can write as Dy = f (x₀ + Dx) – f (x₀).  The ratio Dy/Dx = (change in y)/(change in x) is the average rate of change of the function in the interval between x₀ and x₀ + Dx and is equal to the slope of a line between the two points on a graph.

 

The Derivative:  The derivative of a function  y = f (x) with respect to x is defined as  provided the limit exists. This is the instantaneous rate of change of y with respect to x. The derivative of  y = f (x) with respect to x may be indicated by any one of the following symbols: ,   ,   ,  y’,  f’(x), or

Differentiation:  The process of differentiation is finding the derivative of a function and possibly evaluating the function at a certain value of x. Differentiation is done following a series of formulas rather than the time-consuming limit process of the derivative definition. These formulas are found in many textbooks and handbooks; the most commonly used are printed below. In these formulas u, v, and w are functions of x.

1.       , where c is any constant

2.     

3.     

4.     

5.     

6.     

7.     

8.     

9.     

10. 

The Chain Rule: If u is a function of x and y is a function of u, then y is actually a function of x : y = f(u) , u = g(x) , then y =f{g(x)}.  To find the derivative dy/dx you can substitute one function into the other and solve for y in terms of x and find the derivative. Often it is easier to use the chain rule which states:

Example: If y = u² + 3 and u = 2x + 1, then dy/du =2u, du/dx = 2, and dy/dx = 4u = 8x + 4

Higher Derivatives: Sometimes it is necessary to take a derivative of a derivative, which is called the second derivative. It is designated by one of the symbols ,  y’’, or f’’’(x).  In turn, the derivative of the second derivative is called the third derivative with analogous symbols.