4 Differentiation of a general function from first principles

Consider the graph of y = f ( x ) shown in Figure 7.

Figure 7 :

{ As h rightarrow 0 the chord AB becomes the tangent at A}

Carefully make the following observations:

  1. Point A has coordinates ( x , f ( x ) ) .
  2. Point B has coordinates ( x + h , f ( x + h ) ) .
  3. The straight line A B has gradient
    f ( x + h ) f ( x ) h
  4. If we let h 0 we can find the gradient of the graph of y = f ( x ) at the arbitrary

    point A , provided we can evaluate the appropriate limit on h . The resulting limit is the

    derivative of f with respect to x and is written d f d x or f ( x ) .

    Key Point 3

    Definition of Derivative

    Given y = f ( x ) , its derivative is defined as

    d f d x = f ( x + h ) f ( x ) h  in the limit as  h  tends to 0.
    This is written
    d f d x = lim h 0 f ( x + h ) f ( x ) h

    In a graphical context, the value of d f d x at A is equal to tan θ which is the tangent of the angle that the gradient line makes with the positive x -axis.

Example 1

Differentiate f ( x ) = x 2 + 2 x + 3 from first principles.

Solution

d f d x = lim h 0 f ( x + h ) f ( x ) h

= lim h 0 [ ( x + h ) 2 + 2 ( x + h ) + 3 ] [ x 2 + 2 x + 3 ] h

= lim h 0 [ x 2 + 2 x h + h 2 + 2 x + 2 h + 3 x 2 2 x 3 ] h

= lim h 0 2 x h + h 2 + 2 h h

= lim h 0 2 x + h + 2

= 2 x + 2

Exercises
  1. Use the definition of the derivative to find d f d x when
    1. f ( x ) = 4 x 2 ,
    2. f ( x ) = 2 x 3 ,
    3. f ( x ) = 7 x + 3 ,
    4. f ( x ) = 1 x . (Harder: try
    5. f ( x ) = sin x and use the small angle approximation sin θ θ if θ is small and measured in radians.)
  2. Using your results from Exercise 1 calculate the gradient of the following graphs at the given points:
    1. f ( x ) = 4 x 2 at x = 2 ,
    2. f ( x ) = 2 x 3 at x = 2 ,
    3. f ( x ) = 7 x + 3 at x = 5 ,

       

    4. f ( x ) = 1 x at x = 1 2 .
  3. Find the rate of change of the function y ( x ) = x x + 3 at x = 3 by considering the interval

      x = 3 to x = 3 + h .

    1. 8 x ,  
    2. 6 x 2 ,  
    3. 7 ,  
    4. 1 x 2 ,  
    5. cos x .
    1. 16 ,  
    2. 24,  
    3. 7,  
    4. 4
  1. 1 12