Find the derivative of the function \( G(x)=2\left(x^{4}-\pi x^{2}+6.7\right)^{14} \)

To find the derivative of the function \( G(x) = 2\left(x^{4}-\pi x^{2}+6.7\right)^{14} \), we will apply the chain rule. First, let \( u = x^{4} - \pi x^{2} + 6.7 \). Then, our function can be rewritten as \( G(x) = 2u^{14} \). Now, we find the derivative \( G'(x) \): 1. Differentiate \( G \) with respect to \( u \): \[ \frac{dG}{du} = 2 \cdot 14u^{13} = 28u^{13} \] 2. Now, differentiate \( u \) with respect to \( x \): \[ \frac{du}{dx} = 4x^{3} - 2\pi x \] 3. Now, apply the chain rule: \[ G'(x) = \frac{dG}{du} \cdot \frac{du}{dx} \] \[ G'(x) = 28u^{13}(4x^{3} - 2\pi x) \] 4. Substitute \( u \) back: \[ G'(x) = 28(x^{4} - \pi x^{2} + 6.7)^{13}(4x^{3} - 2\pi x) \] Thus, the derivative of the function is: \[ G'(x) = 28(x^{4} - \pi x^{2} + 6.7)^{13}(4x^{3} - 2\pi x) \]