 Derivatives of Variables      Custom Search   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 n-1=4 so that given and substituting values for n, find that EXAMPLE: Find the derivative of SOLUTION Apply Theorem 2, such that, Therefore, and n-1=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 5-1.-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 5-1 should further the understanding of this section. PRACTICE PROBLEMS: Find the derivatives of the following: ANSWERS:  Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business