Question

\(\frac{d}{dx} \left( \cos(\pi x + \sin \pi x) \right) \)

• A. \( - \sin(\pi x + \sin \pi x) \)
• B. \( - \pi \sin(\pi x) \)
• C. \( - \sin \pi x \)
• D. \( \sin x \)

Hint:

Use the chain rule when differentiating composite functions, applying it separately to both the outer and inner functions.

Answer 1

The Correct Option is B

We are asked to differentiate: \[ \frac{d}{dx} \left( \cos(\pi x + \sin \pi x) \right) \] Use the chain rule for differentiation. First, differentiate the outer function: \[ \frac{d}{dx} \cos(u) = -\sin(u) \] where \( u = \pi x + \sin \pi x \). Next, differentiate the inner function \( u \): \[ \frac{d}{dx} (\pi x + \sin \pi x) = \pi + \cos(\pi x) \cdot \pi \] Thus, the final derivative is: \[ \frac{d}{dx} \left( \cos(\pi x + \sin \pi x) \right) = -\sin(\pi x + \sin \pi x) \cdot \pi \] Thus, the correct answer is \( - \pi \sin(\pi x) \).