Final answer:
The derivative of the function f(x) = 8(7 − 6x)−5/4 is found using the chain rule, resulting in f '(x) = 60(7 − 6x)−9/4, which corresponds to Option C.
Step-by-step explanation:
To find the derivative of the function f(x) = 8(7 − 6x)−5/4, we apply the chain rule of differentiation. The chain rule states that if we have a composite function g(f(x)), then its derivative g'(f(x)) is the product of the derivative of the outside function g' evaluated at f(x) and the derivative of the inside function f'. In this case, our outside function is the exponentiation by −5/4 and our inside function is (7 − 6x).
To differentiate (7 − 6x)−5/4, we first differentiate the inside to get −6, and then we multiply this by the derivative of the outside function. Using the power rule, we bring down the exponent −5/4 to multiply by the coefficient (which in this case is 8), and then subtract one from the exponent. This results in −(5/4) × 8 × −6(7 − 6x)−(5/4) − 1, or simplifying, f '(x) = −(-10)(7 − 6x)−9/4. So, the correct choice is:
Option C: f '(x) = 60(7 − 6x)−9/4