1. **Problem 1: Find the derivative of** $y=\cos(\sin(x))$. 2. Let $f(u)=\cos(u)$ and $g(x)=\sin(x)$ so that $y=f(g(x))$. 3. Using the chain rule: $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$ 4. Calculate derivatives: - $f'(u) = -\sin(u)$ - $g'(x) = \cos(x)$ 5. Substitute back: $$\frac{dy}{dx} = -\sin(\sin(x)) \cdot \cos(x)$$ --- 6. **Problem 2: Find the derivative of** $y = e^{9\sqrt{x}}$. 7. Let $f(u) = e^{u}$ and $g(x) = 9\sqrt{x} = 9x^{1/2}$ so that $y = f(g(x))$. 8. Using the chain rule: $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$ 9. Calculate derivatives: - $f'(u) = e^{u}$ - $g'(x) = 9 \cdot \frac{1}{2} x^{-1/2} = \frac{9}{2\sqrt{x}}$ 10. Substitute back: $$\frac{dy}{dx} = e^{9\sqrt{x}} \cdot \frac{9}{2\sqrt{x}} = \frac{9}{2\sqrt{x}} e^{9\sqrt{x}}$$