b) Use an appropriate method of differentiation to determine the derivative of the following functions (sim- plify your answers as far as possible): (i) \( f(x)=\cos (\sqrt{\sin (\tan \pi x)}) \)

To find the derivative of the function \( f(x) = \cos \left( \sqrt{\sin (\tan (\pi x))} \right) \), we'll apply the chain rule multiple times, working from the outermost function inward. ### Step-by-Step Differentiation: 1. **Identify the Layers of Composition:** \[ f(x) = \cos(g(x)), \quad \text{where} \quad g(x) = \sqrt{h(x)}, \quad \text{and} \quad h(x) = \sin(k(x)), \quad \text{where} \quad k(x) = \tan(\pi x) \] 2. **Differentiate Each Layer:** - **Differentiate the outermost function \( \cos(g(x)) \):** \[ \frac{d}{dx} \cos(g(x)) = -\sin(g(x)) \cdot g'(x) \] - **Differentiate \( g(x) = \sqrt{h(x)} \):** \[ g'(x) = \frac{1}{2} \cdot h(x)^{-1/2} \cdot h'(x) \] - **Differentiate \( h(x) = \sin(k(x)) \):** \[ h'(x) = \cos(k(x)) \cdot k'(x) \] - **Differentiate \( k(x) = \tan(\pi x) \):** \[ k'(x) = \pi \cdot \sec^2(\pi x) \] 3. **Combine the Derivatives:** Putting it all together: \[ f'(x) = -\sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cdot \frac{1}{2} \cdot \left( \sin(\tan(\pi x)) \right)^{-1/2} \cdot \cos(\tan(\pi x)) \cdot \pi \sec^2(\pi x) \] 4. **Simplify the Expression:** \[ f'(x) = -\frac{\pi}{2} \cdot \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cdot \frac{\cos(\tan(\pi x)) \cdot \sec^2(\pi x)}{\sqrt{\sin(\tan(\pi x))}} \] Alternatively, since \( \sec^2(\pi x) = \frac{1}{\cos^2(\pi x)} \), you can write: \[ f'(x) = -\frac{\pi}{2} \cdot \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cdot \frac{\cos(\tan(\pi x))}{\cos^2(\pi x) \cdot \sqrt{\sin(\tan(\pi x))}} \] ### Final Answer: After simplifying, the derivative is –½ π sin √sin(tan πx) multiplied by cos(tan πx) and by sec²πx, all divided by √sin(tan πx). Thus, f ′(x) = −½π sin(√(sin(tan(πx))) ) cos(tan πx) sec²(πx) ⁄ √(sin(tan πx))