Question

Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]

Options

• \[\frac{d}{dx}\left[e^xf(x)\right]=e^x\left[f(x)-f'(x)\right]\]

• \[\frac{d}{dx}\left[e^xf(x)\right]=f(x)+f'(x)\]

• \[\frac{d}{dx}\left[e^xf(x)\right]=e^xf'(x)\]

• \[\frac{d}{dx}\left[e^xf(x)\right]=e^xf(x)+e^xf'(x)=e^x\left[f(x)+f'(x)\right]\]

Answer 1

Apply the product rule to \(e^xf(x)\). The derivative is \(e^xf(x)+e^xf'(x)\), which factors as \(e^x[f(x)+f'(x)]\).