Final answer:
The differentiation of the function g(x) = (x^{-5} + 3)(x^{-3} + 5) is done using the product rule. After finding the derivatives of both parts of the product using the power rule, we combine them in the format of the product rule's formula without further simplification.
Step-by-step explanation:
The question asks us to differentiate the function g(x) = (x-5 + 3)(x-3 + 5). We can apply the product rule of differentiation here, which is used when differentiating products of two functions. This rule states that the derivative of a function u(x)v(x) is u'(x)v(x) + u(x)v'(x), where u'(x) and v'(x) are the derivatives of u(x) and v(x), respectively.
For our function g(x), we define u(x) = x-5 + 3 and v(x) = x-3 + 5. Using the power rule of differentiation, which tells us that the derivative of xn is nxn-1, we can find the derivatives u'(x) and v'(x).
u'(x) = -5x-6 (differentiating x-5)
v'(x) = -3x-4 (differentiating x-3)
Therefore, applying the product rule to g(x) we get:
g'(x) = u'(x)v(x) + u(x)v'(x)
= (-5x-6)(x-3 + 5) + (x-5 + 3)(-3x-4)
Since we were instructed not to simplify, this would be the final form of the differentiated function.