Question

The Fourier Transform of \[ \frac{d}{dt}\left(e^{-at}u(t)\right) \] is

• A. \(\delta(t)\)
• B. \(-ae^{-at}u(t)\)
• C. \(\dfrac{a}{a+j\omega}\)
• D. \(\dfrac{j\omega}{a+j\omega}\)

Hint:

Differentiation in time domain corresponds to multiplication by \(j\omega\) in frequency domain.

Answer 1

The Correct Option is D

Concept: The differentiation property of Fourier Transform states: \[ \frac{d}{dt}x(t) \;\xleftrightarrow{\mathcal F}\; j\omega X(j\omega). \]

Step 1: Find the transform of \(e^{-at}u(t)\).
\[ e^{-at}u(t) \leftrightarrow \frac{1}{a+j\omega}. \]

Step 2: Apply differentiation property.
\[ \frac{d}{dt} \left(e^{-at}u(t)\right) \leftrightarrow j\omega \left( \frac{1}{a+j\omega} \right). \] \[ = \frac{j\omega}{a+j\omega}. \]

Step 3: Write the answer.
\[ \boxed{ \frac{j\omega}{a+j\omega} } \]