Final answer:
To find the derivative of the function g(x) = 1/x^7 using the extended power rule, we rewrite it as g(x) = x^-7, then apply the power rule to obtain the derivative, g'(x) = -7x^-8, which simplifies to g'(x) = -7/x^8.
Step-by-step explanation:
Finding the Derivative Using the Extended Power Rule
To find the derivative of the function g(x) = 1/x7, we can apply the extended power rule for derivatives. The function can be rewritten using a negative exponent as g(x) = x-7. By applying the power rule, we take the exponent, multiply it by the coefficient (which is 1 in this case), and then subtract one from the exponent to derive the new power. This gives us the derivative:
g'(x) = -7x-7-1 = -7x-8
Exponent rules tell us that negative exponents flip the construction to the denominator, so we can write the final derivative as:
g'(x) = -7/x8