Final answer:
To determine h'(x), we apply the chain rule to differentiate h(x) = f²(x) - g²(x), using the given derivatives of f and g. After applying the rule and simplifying, we find that h'(x) = 0 because the product of f and g is constant, leading to the result that the rate of change is zero.
Step-by-step explanation:
To find the derivative of the function h(x) = f²(x) - g²(x), we apply the chain rule and the given information that f'(x) = -g(x) and g'(x) = f(x). Since h'(x) is the derivative of h, we need to differentiate each term separately.
For f²(x), the derivative is 2f(x)f'(x) by the chain rule, which is 2f(x)(-g(x)). For -g²(x), the derivative is -2g(x)g'(x), which equals -2g(x)f(x). When we subtract the derivative of the second term from the first, we have:
h'(x) = 2f(x)(-g(x)) - (-2g(x)f(x))
By simplifying, we find that the terms cancel each other out and we get:
h'(x) = 0
This conclusion is consistent with the given that the product of f and g is a constant. Since their product is constant, their derivatives multiplied by each other would equate to zero rate of change.