1. Problem: Find the derivative $\frac{dy}{dx}$ of $y = 2x^5 - \cos x$. 2. Formula: Use the power rule $\frac{d}{dx} x^n = nx^{n-1}$ and the derivative of cosine $\frac{d}{dx} \cos x = -\sin x$. 3. Step-by-step: 1. Differentiate $2x^5$: $\frac{d}{dx} 2x^5 = 2 \times 5x^{4} = 10x^{4}$. 2. Differentiate $-\cos x$: $\frac{d}{dx} (-\cos x) = -(-\sin x) = \sin x$. 3. Combine results: $\frac{dy}{dx} = 10x^{4} + \sin x$. 4. Explanation: We apply the power rule to the polynomial term and the known derivative of cosine to the trigonometric term, then sum the derivatives. Final answer: $$\frac{dy}{dx} = 10x^{4} + \sin x$$