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The Definition of a Derivative:

The essence of calculus is the derivative. The derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point.

Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit

provided this limit exists.

If this limit exists for each x in an open interval I, then we say that f is differentiable on I. This is also referred to as the "Difference Quotient"

 

 


Example:

Problem:
Consider the function

f(x) = 5x

and the tangent line at x = 0. Zoom in until the graphs of both appear to coincide and estimate the slope of the tangent line from the graph of f.

Visualization:


[Press here to see animation again!]

The slope of the line is approximately

(1.0405 - 0.96282)/(0.024571 + 0.02303) = 1.63

The slope of the tangent line can be shown to be ln( 5) = 1.609...

  • Notation

    We have used the notation f' to denote the derivative of the function f. There are also many other ways to denote the derivative.

    • Df and Dxf are used by some authors to denote the derivative of the function f.
    • If we consider y = f(x), then y' denotes the derivative of the function f.
    • If we let Dx = h and Dy = f(x + h) - f(x) then

      denotes the derivative of the function f.

    • A variation of the previous is

  • The following theorem relates the two properties of differentiability and continuity.

    Theorem. If the function f is differentiable on the interval I. then f is continuous on I except that the reverse is not always true.

    Theorem E. (Power Rule) If f(x) = xn where n is all real numbers, then

    f '(x) = n xn - 1
    .

    •  

     

    Instantaneous Rate of Change of f(x) at x = a: The Derivative

    The instantaneous rate of change of f(x) at x = a is defined by taking the limit of the average rates of change of f over the intervals [a, a+h], as h approaches 0. We write:

      Instantaneous rate of change = lim
      h0
      f(a+h) - f(a)

      h
      		 .
    (We read this as "the limit as h approaches 0 of the difference quotient.".) We also call the instantaneous rate of change the derivative of f at x = a, which we write as f'(a) (read "f prime of a"). Thus,

      f'(a) = lim
      h0
      f(a+h) - f(a)

      h

    Units: The units of f' are units of f per unit of x.

    Quick Example

    If f(x) = 7,500 + 500x - 100x2, then the calculation you did above above suggests (correctly) that

      f'(1) = lim
      h0
      f(1+h) - f(1)

      h
      		 = 300 rupiahs per day

    The Many Rules of Differentiation:

    Product Rule
    Theorem. Let f and g be differentiable functions. Then the derivative of the product fg is

    (fg) '(x) = f(x) g '(x) + g(x) f '(x)

    Quotient Rule
    Theorem. Let f and g be differentiable functions. Then the derivative of the quotient f/g is

    This Rule is also sometimes known as "Lo-De-Hi minus Hi-De-Lo over Lo Squared

     
    Chain Rule
    Theorem. Let F be the composition of two differentiable functions f and g; F(x) = f(g(x)). Then F is differentiable and

    F'(x) = f '(g(x)) g '(x)

     

     
     

     

    Simple Derivatives to Remember...

     

    f(x)

    f'(x)

    Limits

    x 1 lim(x->0) sin(x)/x=1
    a^x a^x · ln(a)  
    x^2 2x lim(x->0) (cos(x)-1)/x=0
    3x^3 9x^2  
    x^2+x 2x+1 lim(x->0) sin(ax)/x=a
    1/x -1/x^2  
    ln(x) 1/x lim(x->0) sin(ax)/sin(bx)=a/b
    ln(g(x)) (g'(x)/g(x))  
    e^g(x) e^g(x) · g'(x) lim(x->0) tan(ax)/x=a
    e^x e^x  
    sin(x) cos(x)  
    cos(x) -sin(x)  
    tan(x) sec^2(x) = 1/cos^2(x)  
    csc(x) -csc(x)cot(x)  
    sec(x) sec(x)tan(x)  
    cot(x) -csc^2(x)  
    arcsin(x) 1/(1-x^2)^½  
    arctan(x) 1/1+x^2  
    arcsec(x) 1/|x|(x^2-1)^½  
      <<

     

     



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