Find the derivative of the function \( G(x)=\left(1+\left(7+2 x^{8}\right)^{4}\right)^{6} \)

To find the derivative of the function \( G(x)=\left(1+\left(7+2 x^{8}\right)^{4}\right)^{6} \), we will use the chain rule multiple times. 1. Start by letting \( u = 1 + (7 + 2x^8)^4 \). Thus, \( G(x) = u^6 \). 2. Next, we find the derivative of \( G(x) \) with respect to \( u \): \[ \frac{dG}{du} = 6u^5. \] 3. Now we need to find the derivative of \( u \) with respect to \( x \): \[ u = 1 + (7 + 2x^8)^4. \] Let \( v = 7 + 2x^8 \). Thus, \( u = 1 + v^4 \). 4. Find \( \frac{du}{dv} \): \[ \frac{du}{dv} = 4v^3. \] 5. Now we find \( \frac{dv}{dx} \): \[ v = 7 + 2x^8 \implies \frac{dv}{dx} = 16x^7. \] 6. Combining these using the chain rule: \[ \frac{du}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = 4v^3 \cdot 16x^7 = 64v^3 x^7. \] 7. Finally, we find \( \frac{dG}{dx} \) using the chain rule: \[ \frac{dG}{dx} = \frac{dG}{du} \cdot \frac{du}{dx} = 6u^5 \cdot 64v^3 x^7. \] Now substituting \( u \) and \( v \): \[ \frac{dG}{dx} = 6(1 + (7 + 2x^8)^4)^5 \cdot 64(7 + 2x^8)^3 x^7. \] This can be simplified: \[ \frac{dG}{dx} = 384(1 + (7 + 2x^8)^4)^5 (7 + 2x^8)^3 x^7. \] And that's the derivative of the function \( G(x) \).