Differentiate the function \( k(x)=3 \pi x \sin x+4 \pi x \). Write the exact answer. Do not round.

To differentiate the function \( k(x) = 3 \pi x \sin x + 4 \pi x \), we can use the product rule for the term \( 3 \pi x \sin x \) and differentiate the constant times \( x \) directly. Using the product rule on \( 3 \pi x \sin x \): Let \( u = 3 \pi x \) and \( v = \sin x \). Then, \[ \frac{du}{dx} = 3 \pi \quad \text{and} \quad \frac{dv}{dx} = \cos x. \] Applying the product rule: \[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} = (3 \pi x)(\cos x) + (\sin x)(3 \pi). \] Now, differentiating \( 4 \pi x \): \[ \frac{d}{dx}(4 \pi x) = 4 \pi. \] Combining these results, we get: \[ k'(x) = (3 \pi x \cos x) + (3 \pi \sin x) + 4 \pi. \] Thus, the exact answer is: \[ k'(x) = 3 \pi x \cos x + 3 \pi \sin x + 4 \pi. \]