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 \( f(x) = \cos(\sqrt{\sin(\tan(\pi x))}) \), you’ll want to apply the chain rule multiple times. First, let’s identify the outermost function, which is the cosine function. The derivative of \( \cos(u) \) is \( -\sin(u) \cdot \frac{du}{dx} \), where \( u = \sqrt{\sin(\tan(\pi x))} \). Next, differentiate \( u \): 1. The derivative of \( \sqrt{v} \) (where \( v = \sin(\tan(\pi x)) \)) is \( \frac{1}{2\sqrt{v}} \cdot \frac{dv}{dx} \). 2. The derivative of \( v = \sin(w) \) (where \( w = \tan(\pi x) \)) is \( \cos(w) \cdot \frac{dw}{dx} \). 3. Finally, for \( w = \tan(\pi x) \), the derivative is \( \sec^2(\pi x) \cdot \frac{d(\pi x)}{dx} = \pi \sec^2(\pi x) \). Putting it all together using the chain rule results in: \[ f'(x) = -\sin(\sqrt{\sin(\tan(\pi x))}) \cdot \frac{1}{2\sqrt{\sin(\tan(\pi x))}} \cdot \cos(\tan(\pi x)) \cdot \pi \sec^2(\pi x) \] This compiles to give a complex expression that succinctly represents the interactions of the three nested functions. Now remember, as you tackle derivatives of complex functions, keeping track of the chain rule and applying it piece by piece can seem daunting but leads to beautiful simplifications! Happy differentiating!