1. **State the problem:** We need to find the derivative $\frac{dy}{dx}$ of the function $y = 10^{1 - x^2}$. 2. **Recall the formula:** For a function of the form $y = a^{u(x)}$, the derivative is given by $$\frac{dy}{dx} = a^{u(x)} \cdot \ln(a) \cdot \frac{du}{dx}$$ where $a$ is a positive constant and $u(x)$ is a function of $x$. 3. **Identify components:** Here, $a = 10$ and $u(x) = 1 - x^2$. 4. **Compute $\frac{du}{dx}$:** $$\frac{du}{dx} = \frac{d}{dx}(1 - x^2) = 0 - 2x = -2x$$ 5. **Apply the formula:** $$\frac{dy}{dx} = 10^{1 - x^2} \cdot \ln(10) \cdot (-2x) = -2x \ln(10) \cdot 10^{1 - x^2}$$ 6. **Final answer:** $$\boxed{\frac{dy}{dx} = -2x \ln(10) \cdot 10^{1 - x^2}}$$