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DERIVATIVES OF VARIABLES In this section of variables, we will extend the theorems
of limits covered previously. Recall that a derivative is actually a limit. The
proof of the theorems presented here involve the delta process. POWER FORM Theorem 2. The derivative of the function
is
given by
if
n is any real number. PROOF: By definition
The expression
may be expanded by the binomial theorem into
Substituting in the expression for the derivative, we have _{} Simplifying, this becomes _{} Letting
x
approach zero, we have
Thus, the proof is complete. EXAMPLE: Find the derivative of
SOLUTION: Apply Theorem 2, such that, _{
} Therefore, n=5 and n1=4 so that given
and substituting values for n, find
that
EXAMPLE: Find the derivative of
SOLUTION Apply Theorem 2, such that,
Therefore,
and n1=0 so that
The previous example is a special case of the power form
and indicates that the derivative of a function with respect to itself is 1. EXAMPLE: Find the derivative of
where a is a constant. SOLUTION
and
so that
Therefore,
Table
51.Derivatives of Functions
The previous example is a continuation of the derivative
of a function with respect to itself and indicates that the derivative of a
function with respect to itself, times a constant, is that constant. EXAMPLE. Find the derivative of
SOLUTION: ^{
} A study of the functions and their derivatives in table 51 should
further the understanding of this section. PRACTICE PROBLEMS: Find the derivatives of the following:
ANSWERS:
