Introducing Differentiation

(1) Definition (Derivative) The derivative of a function for any is given by



provided this limit exists. The derivative is also denoted by and other common notations are , , and Also the limit is sometimes denoted by



The process of finding the derivative is called differentiation. A function is differentiable at when the defining limit exists; and we say that is differentiable on when is differentiable at every point in

(2) Example (Derivative) Find the derivative function using the definition of the derivative. Find the derivative of the function at

    Solution. By definition we compute the limit, as follows,
    










which is the derivative of for any At we have

(3) Example (Derivative) Find the derivative function using the definition of the derivative. Find the derivative of the function at

    Solution. By definition we compute the limit, as follows,
    












which is the derivative of for any At we have (Notice that we made use of the formula ).

(4) Proposition (Tangent Line) The slope of a tangent line to the function at is given by   and the equation of the tangent line is given by

(5) Example (Tangent Line) Find the equation of the tangent line at the indicated point. Find the equation of the tangent line to the graph of at

    Solution. To find the slope of the tangent line we compute the derivative of the function,
    











    


which is the derivative of the function for any At we have



Therefore, the equation of the tangent line at is





As an illustration the graph of the given function and the tangent line at is

(6) Example (Tangent Line) Find the equation of the tangent line at the indicated point. Find the equation of the tangent line to the graph of at

    Solution. To find the slope of the tangent line we compute the derivative of the function,
    












which is the derivative of the function for any At we have



Therefore, the equation of the tangent line at is





As an illustration the graph of the given function and the tangent line at is

(7) Example (Existence of Derivative) Give three examples of functions where is not differentiable at but is defined at   

    Solution. The function is not differentiable at since





which proves the two-sided limit (the derivative) does not exist. This type of example where the function is not differentiable is called a corner point.
    Secondly, the function   is not differentiable at since





which proves the two-sided limit (the derivative) does not exist. This type of example where the function is not differentiable is called a vertical tangent.
    Thirdly,  the function is not differentiable at since





which proves the two-sided limit (the derivative) does not exist.

(8) Example (Existence of Derivative)  Compute the difference quotient for the function defined by



Do you think is differentiable at ? If so, what is the equation of the tangent line at ?

    Solution.  For we find,



At we have, Using a table of values to compute the limit we infer that and so the equation of the tangent line is

(9) Proposition (Differentiability Implies Continuity) If a function is differentiable at then it is also continuous at .

(10) Proof. Assume that is a differentiable function, then by definition,



exists, and therefore we can use the product rule for limits with,











Whence, and so is continuous at

(11) Example (Differentiability Implies Continuity) The following graphs illustrate the relationship between continuity and differentiability.



For example, the graph of the left is continuous at every real number but this graph is not differentiable at The graph of the right is not continuous at so it certainty is not differentiable there.

• print this page
• pages linking here
• related pages

Cite this as:
Introducing Differentiation
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/introducing-differentiation.html