b) Use an appropriate method of differentiation to determine the derivative of the following functions (sim-
plify your answers as far as possible):
(i)

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Jan 23,2025

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

UpStudy AI · Accepted Answer

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))
The derivative of $$ f(x) = \cos \left( \sqrt{\sin (\tan (\pi x))} \right) $$ is: $$ f'(x) = -\frac{\pi}{2} \cdot \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cdot \frac{\cos(\tan(\pi x)) \cdot \sec^2(\pi x)}{\sqrt{\sin(\tan(\pi x))}} $$

b) Use an appropriate method of differentiation to determine the derivative of the following functions (sim-
plify your answers as far as possible):
(i)

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