1. **State the problem:** Find the derivative of the function $$f(x) = 2^x \log_3 \left(7^{x^2 - 4}\right)$$. 2. **Rewrite the function:** Use the logarithm power rule $$\log_a b^c = c \log_a b$$ to simplify: $$f(x) = 2^x (x^2 - 4) \log_3 7$$. 3. **Identify constants:** Note that $$\log_3 7$$ is a constant. 4. **Apply the product rule:** The function is a product of two functions: $$u = 2^x$$ and $$v = (x^2 - 4) \log_3 7$$. The product rule states: $$f'(x) = u'v + uv'$$. 5. **Find derivatives:** - Derivative of $$u = 2^x$$ is $$u' = 2^x \ln 2$$. - Derivative of $$v = (x^2 - 4) \log_3 7$$ is $$v' = 2x \log_3 7$$ since $$\log_3 7$$ is constant. 6. **Combine results:** $$f'(x) = 2^x \ln 2 \cdot (x^2 - 4) \log_3 7 + 2^x \cdot 2x \log_3 7$$. 7. **Factor common terms:** $$f'(x) = 2^x \log_3 7 \left[(x^2 - 4) \ln 2 + 2x\right]$$. **Final answer:** $$\boxed{f'(x) = 2^x \log_3 7 \left[(x^2 - 4) \ln 2 + 2x\right]}$$