Derivatives Using the Limit Definition

DERIVATIVES USING THE LIMIT DEFINITION

The following problems require the use of the limit definition of a derivative, which is given by

.

They range in difficulty from easy to somewhat challenging. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by making proper use of functional notation and careful use of basic algebra. Keep in mind that the goal (in most cases) of these types of problems is to be able to divide out the term so that the indeterminant form of the expression can be circumvented and the limit can be calculated.

SOLUTIONS TO DERIVATIVES USING THE LIMIT DEFINITION

SOLUTION 1 :

(Algebraically and arithmetically simplify the expression in the numerator.)

(The term now divides out and the limit can be calculated.)

.

SOLUTION 2 :

(Algebraically and arithmetically simplify the expression in the numerator.)

(Factor from the expression in the numerator.)

(The term now divides out and the limit can be calculated.)

.

SOLUTION 3 :

(Eliminate the square root terms in the numerator of the expression by multiplying

by the conjugate of the numerator divided by itself.)

(Recall that )

(The term now divides out and the limit can be calculated.)

.

SOLUTION 4 :

(Get a common denominator for the expression in the numerator. Recall that division by is the same as multiplication by . )

(Algebraically and arithmetically simplify the expression in the numerator. It is important to note that the denominator of this expression should be left in factored form so that the term can be easily eliminated later.)

(The term now divides out and the limit can be calculated.)

.

SOLUTION 5 :

(At this point it may appear that multiplying by the conjugate of the numerator over

itself is a good next step. However, doing something else is a better idea.)

(Note that A - B can be written as the difference of cubes , so that

. This will help explain the next step.)

(Algebraically and arithmetically simplify the expression in the numerator.)

(The term now divides out and the limit can be calculated.)

.

SOLUTION 6 :

(Recall a well-known trigonometry identity :

.)

(Recall the following two well-known trigonometry limits :

and .)

.

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