Techniques of Differentiation

(1) Proposition (Differentiation Formulas) Let be a function.

    (i) If is a constant function, for any real number then
    
    (ii) If is a power function, for any real number , then

    (iii) If for any two functions and then

    (iv) If for any two functions and then

    (v) If for any two functions and , and any two constants and , then

    (vi) If for any two functions and , then

    (vii) If for any two functions and , then

(2) Example (Differentiation Formulas) Find the derivative of the following function

    Solution. Since is a constant with respect to , we use the constant rule to find    



(3) Example (Differentiation Formulas) Find the derivative of the following function  

    Solution. Using the power rule, linearity rule, and the sum rule, we find

.


(4) Example (Differentiation Formulas) Find the derivative of the following function

    Solution. We use the product rule with , and We find
    








    
(5) Example (Differentiation Formulas) Find the derivative of the following function  

    Solution.  We use the product rule with , and We find
    






Since







Thus,



which simplifies to,





(6) Example (Differentiation Formulas) Find the derivative of the following function  

    Solution. We use the quotient rule with and But first we compute
    
              and       

Thus,





which simplifies to



or




    
(7) Example (Differentiation Formulas) Find the derivative of the following function

    Solution. Using the product rule with we find



Using the quotient rule with   , , and we find



The second expression for is easier to work with.

(8) Example (Differentiation Formulas) Find the derivative of the following function  

    Solution. We can rewrite as so as to use the power rule to find,

(9) Proposition (Equation of a Tangent Line) If exists then the equation of the tangent line to the curve at the point is

(10) Example (Equation of a Tangent Line) Find the equations of the tangent lines to the curve that are parallel to the line

    Solution. The line has slope and we use this with the derivative  of to find the Since we have Solving for we get and Therefore, the points of tangency are at and   The tangent lines are found by using where with and We find and respectively. Therefore, the equations of the tangent lines are and Here's is a graph of and the tangent lines:




(11) Example (Equation of a Tangent Line) How many tangent lines to the curve pass through the point ? At which points do these tangent lines touch the curve?

    Solution. All tangent lines through have the form where Since we our looking for the intersection (point of tangency) we eliminate as follows:



Solving for we obtain, Thus there are two tangent lines and they are tangent at the point Here's the graph of the two tangent lines through along with





(12) Example (Equation of a Tangent Line) Find the equations of both tangent lines through the point that are tangent to the parabola

    Solution. All tangent lines through have the form where Since we our looking for the intersection (point of tangency) we eliminate as follows:



Solving for we obtain, and Thus there are two tangent lines and they are tangent at the points and The tangent lines are and Here's the graph of the two tangent lines through along with

(13) Proposition (Horizontal Tangent Line) If then the equation of the tangent line to the curve at the point is and is said to have a horizontal tangent line at

(14) Example (Horizontal Tangent Line) For what values of does the graph of have a horizontal tangent?

    Solution. To find the horizontal tangent lines we find where the derivative is 0. We compute, So we need to solve We find,  





And using the quadratic formula we have Thus, the values of 9x0 where the tangents lines are horizontal are
    
(15) Example (Horizontal Tangent Line)  Find the points on the curve where the tangent line is horizontal.

    Solution. To find the horizontal tangent lines we find where the derivative is 0. We compute, So we need to solve Using the quadratic formula we have and Thus, the values of where the tangents lines are horizontal are and   

    If is a differentiable function, then its derivative is also a function, so may have a derivative of its own, denoted by This function is called the second derivative of Moreover, the second derivative may be differentiable, and etc.

(16) Definition (Higher-Order Derivatives) Suppose and are differentiable functions, then the second derivative of is defined as and is denoted by Further, the third derivative is defined as and is denoted by ; and the fourth derivative is defined as and is denoted by , provided these functions exist. In general, if is differentiable, then is the derivative of

    In Leibniz notation the first, second the third derivatives are

              and  

The derivative is denoted by and in Leibniz notation:

(17) Example (Higher-Order Derivatives) Find the first, second, and third derivatives of



    Solution. We could use the product rule but since we want higher order derivatives it will be quicker to expand first. We find,



Thus,









(18) Example (Higher-Order Derivatives) Find the first, second, and third derivatives of
    
    Solution. To find the first derivative we use the quotient rule with and   Since, and we have,
    






Similarly, we use the quotient rule to find the second derivative,





Similarly, we use the quotient rule to find the third derivative,

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