1. **State the problem:** Find the derivative of the function $$y = e^{\sqrt{7x^2 - 1}}$$. 2. **Recall the formula:** The derivative of $$e^{u(x)}$$ with respect to $$x$$ is $$\frac{dy}{dx} = e^{u(x)} \cdot u'(x)$$. 3. **Identify the inner function:** Here, $$u(x) = \sqrt{7x^2 - 1} = (7x^2 - 1)^{1/2}$$. 4. **Differentiate the inner function:** $$ u'(x) = \frac{1}{2}(7x^2 - 1)^{-1/2} \cdot (14x) = \frac{14x}{2\sqrt{7x^2 - 1}} = \frac{7x}{\sqrt{7x^2 - 1}} $$ 5. **Apply the chain rule:** $$ \frac{dy}{dx} = e^{\sqrt{7x^2 - 1}} \cdot \frac{7x}{\sqrt{7x^2 - 1}} $$ 6. **Final answer:** $$ \boxed{\frac{dy}{dx} = \frac{7x}{\sqrt{7x^2 - 1}} e^{\sqrt{7x^2 - 1}}} $$ This derivative uses the chain rule, differentiating the outer exponential function and then multiplying by the derivative of the inner square root function.