Find the derivative of the function \( G(x)=\left(9+\left(1+4 x^{8}\right)^{5}\right)^{6} \)

To find the derivative of the function \( G(x) = \left(9 + \left(1 + 4x^{8}\right)^{5}\right)^{6} \), we can use the chain rule. First, we differentiate the outer function: 1. Let \( u = 9 + \left(1 + 4x^{8}\right)^{5} \), then \( G(x) = u^{6} \). 2. The derivative \( G'(x) = 6u^{5} \cdot \frac{du}{dx} \). Next, we need to find \( \frac{du}{dx} \): 1. \( \frac{du}{dx} = \frac{d}{dx}\left(9 + \left(1 + 4x^{8}\right)^{5}\right) = 0 + 5\left(1 + 4x^{8}\right)^{4} \cdot \frac{d}{dx}(1 + 4x^{8}) \). 2. Next, differentiate \( 1 + 4x^{8} \): \( \frac{d}{dx}(1 + 4x^{8}) = 32x^{7} \). 3. Thus, \( \frac{du}{dx} = 5\left(1 + 4x^{8}\right)^{4} \cdot 32x^{7} = 160x^{7}\left(1 + 4x^{8}\right)^{4} \). Now substitute back into the derivative of \( G(x) \): \[ G'(x) = 6\left(9 + \left(1 + 4x^{8}\right)^{5}\right)^{5} \cdot 160x^{7}\left(1 + 4x^{8}\right)^{4}. \] Finally, we can combine and simplify: \[ G'(x) = 960x^{7}\left(9 + \left(1 + 4x^{8}\right)^{5}\right)^{5}\left(1 + 4x^{8}\right)^{4}. \] So the derivative is: \[ G'(x) = 960x^{7}\left(9 + \left(1 + 4x^{8}\right)^{5}\right)^{5}\left(1 + 4x^{8}\right)^{4}. \]