Introduction

The main idea of derivative is that of rate of change of a function.  The primary mathematical tool that is used to calculate rate of change and the slopes of curves is derivative.  Slope of the graph of y=f(x) at x=x0is given by 

The ratio    is called a difference quotient.

The difference quotient can also be interpreted as the average rate of change of f(x) over the interval [x₀, x₁] and its limit as x₁     x₀ is the instantaneous rate of change of f(x) at x=x₀.

Derivative of a function

Suppose that x₀ is a number in the domain of a function f then the derivative of ‘f’ at x=x₀ and is denoted by f’(x₀) and is defined as
                                       

If the limit of the difference quotient exists, f’(x₀) is the slope of the graph of ‘f’ at the point x=x₀.  If this limit does not exist, then the slope of the graph of ‘f’ is undefined at x=x₀
Example: Find the slope of the function y=x²+1 at the point (2, 5)
Solution: Slope of the curve at the point (2, 5) is given by

Graphical representation of Derivative

Suppose that x₀ is a number in the domain of a function ‘f’. If

then we define the tangent line to the graph of f at the point P(x₀, f(x₀) to be the line whose equation is  y – f(x₀) = f’(x₀)(x-x₀).
Below is the graph of a function f(x) and a rough sketch of f’(x).  The graph of f’(x) represents the slopes of the tangent lines to a point on the graph for each x-value in the domain of f(x). For example if one draws a tangent line through the point (0.5, 3) its slope will be zero.  Looking at the graph of f’(x) at x=0.5, y=0

Example: Find the derivative with respect to x of f(x)=x3-x.  Graph f and f’ together and discuss the relationship between the two graphs

Solution: 
                 

                   
Since f’(x) can be interpreted as the slope of the graph of y=f(x) at x, the derivative f’(x) is positive where the graph of f has positive slope, it is negative where the graph of f has negative slope, and it is zero where the graph of f is horizontal.

Interpretation of the derivative

The derivative f’ of a function f can be interpreted as a function whose value at x is the slope of the graph of y=f(x) at x, or alternatively, it can be interpreted as a function whose value at x is the instantaneous rate of change of y with respect to x at x.  In particular, when y=f(t) describes the position at time t of an object moving along a straight line, then f’(t) describes the position at time t of an object moving along a straight line, then f’(t) describes the instantaneous velocity of the object at time ‘t’.

From the figure at each value of ‘x’, the tangent line to a line y=mx+b coincides with the line itself and hence all tangent lines have slope m.  This suggests geometrically that if f(x) = mx + b, then f’(x) = m for all x. This is confirmed by the following computations:
      

Example: The graph of y=|x| has a corner at x=0 which implies that f(x)=|x| is not differentiable at x=0
a)    Prove that f(x)=|x| is not differentiable at x=0 by showing that the limit does not exist at x=0
b)    Find a formula for f’(x)
                      

Solution: From the definition
           

does not exist because the one sided limits are not equal.
b) A formula for the derivatives of f(x) =|x| can be obtained by writing |x| in piecewise form and treating the cases x>0 and x<0 separately.  If x>0, then f(x) =x and f’(x)=1; if x<0, then f(x)-x and f’(x)=-1.  Thus,
                                              

The graph of f’ is shown below.  We can see that f’ is not continuous at x=0.  This shows that a function that is continuous everywhere may have a derivative that fails to be continuous everywhere
                      

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Reference Links:

• http://en.wikipedia.org/wiki/Derivative
• http://en.wikipedia.org/wiki/Difference_quotient
• http://en.wikipedia.org/wiki/Slope
• http://en.wikipedia.org/wiki/Tangent_lines_to_circles