Question

Find the derivative of \( x^5 + \cos 2x \): \[ \frac{d}{dx} \left( x^5 + \cos 2x \right) \]

• A. \( 5x^4 + \sin 2x \)
• B. \( 5x^4 + \cos 2x \)
• C. \( 5x^4 - 2\sin 2x \)
• D. \( x^5 + 2\sin 2x \)

Hint:

When differentiating a function like \( \cos 2x \), use the chain rule: \( \frac{d}{dx} \left( \cos(g(x)) \right) = -\sin(g(x)) \cdot g'(x) \). Here, \( g(x) = 2x \), so \( g'(x) = 2 \).

Answer 1

The Correct Option is C

We need to differentiate \( x^5 + \cos 2x \) with respect to \( x \). Using basic differentiation rules: \[ \frac{d}{dx} \left( x^5 \right) = 5x^4 \] Next, applying the chain rule to \( \cos 2x \): \[ \frac{d}{dx} \left( \cos 2x \right) = -\sin 2x \cdot 2 = -2\sin 2x \] Thus, the derivative is: \[ 5x^4 - 2\sin 2x \] Therefore, the correct option is: \[ \boxed{5x^4 - 2\sin 2x} \]