Final answer:
The power rule for differentiation can be proved using the derivative quotient rule for negative integers. To make the function continuous at x=0, the value of c should be equal to the limit of the function as x approaches 0, which is 0.
Step-by-step explanation:
To prove the power rule for negative integers using the derivative quotient rule, we need to apply the quotient rule to the function f(x) = 1/x^m, where m is a positive integer. Using the quotient rule, we get:
f'(x) = (x^m * 0 - 1 * (m * x^(m-1))) / (x^m)^2 = -mx^(m-1) / x^2 = -mx^(m-1-2) = -mx^(-m-1).
Therefore, d/dx(x^(-m)) = -mx^(-m-1), proving the power rule for negative integers.
To find a value of c that will make f(x) continuous at x=0, we need to find the limit of f(x) as x approaches 0. By evaluating the limit, we can determine the value of c. Since the numerator, sin^2(3x), approaches 0 as x approaches 0, to make f(x) continuous, the value of c should be equal to that limit, which is 0. Therefore, f(x) = 0 for x=0 to make it continuous.
Learn more about power rule for negative integers