Final answer:
To find f'(x), the derivative of f(x)=(3x+8)⁻³, we can use the power rule and the chain rule. Applying the power rule and chain rule, we get f'(x)=-9(3x+8)⁻⁸.
Step-by-step explanation:
To find f'(x), the derivative of f(x)=(3x+8)⁻³, we can use the power rule for differentiation and the chain rule. The power rule states that if we have a function of the form f(x)=xⁿ, the derivative is f'(x)=n*xⁿ⁻¹. Applying the power rule to our function, we get f'(x)=(-3)*(3x+8)⁻⁴. However, since we have a composite function, we need to apply the chain rule as well. The chain rule states that if we have a function of the form g(f(x)), the derivative is g'(f(x))*f'(x). In our case, g(u)=u⁻³ and f(x)=3x+8. Taking the derivative of g(u), we get g'(u)=-3u⁻⁴. Plugging in f(x) for u, we get g'(f(x))=-3(3x+8)⁻⁴. Finally, we multiply g'(f(x)) and f'(x) to get the derivative of the composite function f(x), which is f'(x)=(-3)*(3x+8)⁻⁴*(-3(3x+8)⁻⁴)=-9(3x+8)⁻⁸.