1. **Problem Statement:** Find the derivative $\frac{dy}{dx}$ if $y = e^{\tan^{-1} x}$. 2. **Formula and Rules:** To find $\frac{dy}{dx}$ when $y$ is a function of $x$, use the chain rule: $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$ where $u = \tan^{-1} x$. 3. **Step 1: Identify inner and outer functions:** - Outer function: $e^u$ where $u = \tan^{-1} x$. - Inner function: $u = \tan^{-1} x$. 4. **Step 2: Differentiate outer function with respect to $u$:** $$\frac{dy}{du} = e^u$$ 5. **Step 3: Differentiate inner function with respect to $x$:** The derivative of $\tan^{-1} x$ is $$\frac{du}{dx} = \frac{1}{1 + x^2}$$ 6. **Step 4: Apply chain rule:** $$\frac{dy}{dx} = e^u \times \frac{1}{1 + x^2}$$ Substitute back $u = \tan^{-1} x$: $$\frac{dy}{dx} = e^{\tan^{-1} x} \times \frac{1}{1 + x^2}$$ 7. **Step 5: Final answer:** $$\boxed{\frac{dy}{dx} = \frac{e^{\tan^{-1} x}}{1 + x^2}}$$ **Explanation for a 5-year-old:** Imagine you have a magic box that changes $x$ into $\tan^{-1} x$, and then another magic box that changes that into $e^{\text{that}}$. To find how fast the final number changes when $x$ changes, you find how fast the second box changes with its input, and multiply by how fast the first box changes with $x$. That's the chain rule! **Tip:** When you see a function inside another function, always think about the chain rule. Differentiate the outside first, then multiply by the derivative of the inside. **Answer choice:** (a) $\frac{y}{1 + x^2}$ because $y = e^{\tan^{-1} x}$, so the derivative matches option (a).