Derivative Exponential 6E2F42

1. **State the problem:** We need to find the derivative of the function $$y = e^{ax^9}$$ with respect to $x$. 2. **Recall the formula:** The derivative of an exponential function $e^{u(x)}$ is given by $$\frac{dy}{dx} = e^{u(x)} \cdot u'(x)$$ where $u(x)$ is a function of $x$. 3. **Identify $u(x)$:** Here, $$u(x) = ax^9$$. 4. **Find $u'(x)$:** Differentiate $u(x)$ with respect to $x$: $$u'(x) = \frac{d}{dx}(ax^9) = a \cdot \frac{d}{dx}(x^9) = a \cdot 9x^8 = 9ax^8$$ 5. **Apply the chain rule:** Using the formula, $$\frac{dy}{dx} = e^{ax^9} \cdot 9ax^8$$ 6. **Final answer:** $$\boxed{\frac{dy}{dx} = 9ax^8 e^{ax^9}}$$