Techniques of Differentiation

    Computing the limit of the difference quotient can be tedious and require ingenuity; fortunately for a large number of common function there is a better way to compute the derivative. In this topic, we detail the power rule, product rule and the quotient rule for differentiation. These rules greatly simplify the task of differentiation. We also give examples on how to find the tangent line give some geometric information; and to find the horizontal tangent lines to the graph of a given function.

    The next theorem states the common procedural rules for taking derivatives. For example, the derivative of a sum of functions is the sum of the derivative functions. The same is not true for a product of functions. To convince yourself that the derivative of the product of two functions is not the product of the derivative functions try an example, say and

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

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    



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

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

.


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

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








    
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,





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




    
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.

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

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

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