Use the chain rule to find the derivative of \[ f(x)=4\left(10 x^{3}-2 x^{7}\right)^{18} \] You do not need to expand out your answer. \( f^{\prime}(x)= \)

To find the derivative of the function \[ f(x) = 4\left(10 x^{3} - 2 x^{7}\right)^{18}, \] we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \[ f'(x) = f'(g(x)) \cdot g'(x). \] In this case, we can identify: - The outer function \( f(u) = 4u^{18} \) where \( u = 10x^3 - 2x^7 \). - The inner function \( g(x) = 10x^3 - 2x^7 \). First, we need to find the derivative of the outer function \( f(u) \): \[ f'(u) = 4 \cdot 18u^{17} = 72u^{17}. \] Next, we find the derivative of the inner function \( g(x) \): \[ g'(x) = \frac{d}{dx}(10x^3) - \frac{d}{dx}(2x^7) = 30x^2 - 14x^6. \] Now, we can apply the chain rule: \[ f'(x) = f'(g(x)) \cdot g'(x) = 72(10x^3 - 2x^7)^{17} \cdot (30x^2 - 14x^6). \] Thus, the derivative of \( f(x) \) is \[ f'(x) = 72(10x^3 - 2x^7)^{17} (30x^2 - 14x^6). \] This is the final answer.