Answer
To find the derivative of \( f(x) = \cos\left( \sqrt{ \sin(\tan(\pi x)) } \right) \), use the chain rule multiple times:
1. Differentiate the outer cosine function: \( -\sin(w) \).
2. Differentiate the square root: \( \frac{1}{2\sqrt{v}} \).
3. Differentiate the sine function: \( \cos(u) \).
4. Differentiate the tangent function: \( \pi \sec^2(\pi x) \).
Combine these derivatives:
\[
f'(x) = -\frac{\pi \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cos(\tan(\pi x)) \sec^2(\pi x)}{2 \sqrt{\sin(\tan(\pi x))}}
\]
So, the derivative is:
\[
f'(x) = -\frac{\pi \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cos(\tan(\pi x)) \sec^2(\pi x)}{2 \sqrt{\sin(\tan(\pi x))}}
\]
Solution
To differentiate the function
\[
f(x) = \cos\left( \sqrt{ \sin\left( \tan(\pi x) \right) } \right),
\]
we can apply the **chain rule** multiple times due to the nested functions involved. Here's a step-by-step breakdown:
### Step 1: Identify the Nested Functions
Let's break down \( f(x) \) into simpler functions:
1. **Innermost function:**
\[
u = \tan(\pi x)
\]
2. **Next layer:**
\[
v = \sin(u) = \sin(\tan(\pi x))
\]
3. **Following layer:**
\[
w = \sqrt{v} = \sqrt{\sin(\tan(\pi x))}
\]
4. **Outer function:**
\[
f(x) = \cos(w) = \cos\left( \sqrt{\sin(\tan(\pi x))} \right)
\]
### Step 2: Apply the Chain Rule
To find \( f'(x) \), we'll differentiate each layer step by step.
1. **Differentiate \( f \) with respect to \( w \):**
\[
\frac{df}{dw} = -\sin(w)
\]
2. **Differentiate \( w \) with respect to \( v \):**
\[
\frac{dw}{dv} = \frac{1}{2\sqrt{v}} = \frac{1}{2\sqrt{\sin(\tan(\pi x))}}
\]
3. **Differentiate \( v \) with respect to \( u \):**
\[
\frac{dv}{du} = \cos(u) = \cos(\tan(\pi x))
\]
4. **Differentiate \( u \) with respect to \( x \):**
\[
\frac{du}{dx} = \frac{d}{dx} [\tan(\pi x)] = \pi \sec^2(\pi x)
\]
### Step 3: Combine the Derivatives
Using the chain rule, multiply the derivatives from each layer:
\[
f'(x) = \frac{df}{dw} \cdot \frac{dw}{dv} \cdot \frac{dv}{du} \cdot \frac{du}{dx}
\]
Substituting the derivatives:
\[
f'(x) = \left[ -\sin\left( \sqrt{\sin(\tan(\pi x))} \right) \right] \cdot \left[ \frac{1}{2\sqrt{\sin(\tan(\pi x))}} \right] \cdot \cos(\tan(\pi x)) \cdot \pi \sec^2(\pi x)
\]
### Step 4: Simplify the Expression
Combining the constants and trigonometric functions, the derivative simplifies to:
\[
f'(x) = -\frac{\pi \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cos(\tan(\pi x)) \sec^2(\pi x)}{2 \sqrt{\sin(\tan(\pi x))}}
\]
Alternatively, it can be written as:
\[
f'(x) = -\frac{\pi \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cos(\tan(\pi x))}{2 \cos^2(\pi x) \sqrt{\sin(\tan(\pi x))}}
\]
### Final Answer
Thus, the derivative of \( f(x) \) is:
\[
f'(x) = -\frac{\pi \sin\left( \sqrt{\sin(\tan(\pi x))} \right) \cos(\tan(\pi x)) \sec^2(\pi x)}{2 \sqrt{\sin(\tan(\pi x))}}
\]
### Summary
By applying the chain rule methodically to each nested function, we successfully derived the derivative of the given complex trigonometric function.
Reviewed and approved by the UpStudy tutoring team
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