Question

Which of the following are eigen functions of d/dx?

log(x)

exp(x)

sin(x)

xn​

Answer 1

Answer:

b) exp(x)

Explanation:

For eigen function:

$\\\tt \Omega \psi = \omega \psi$

Where on applying operator $\\\tt \Omega$ on function $\\\tt \psi$ returns a constant $\\\tt \omega$ times the function itself.

Comparing each option one by one:

1. $\dfrac{d( \log x)}{dx} = \dfrac{1}{x}$   # is not.
2. $\\\tt \dfrac{d( e^x)}{dx} = 1 . e^x$ # is
3. $\\\tt \dfrac{d \ sinx }{dx} =cos(x)$ #is not
4. $\\\tt \dfrac{d(xn)}{dx} =n$ # is not.

Answer 2

Answer:

b) exp(x)

Explanation:

For eigen function:

\begin{gathered}\\\tt \text{\O}mega \psi = \omega \psi\end{gathered}

Ømegaψ=ωψ

Where on applying operator \begin{gathered}\\\tt \text{\O}mega\end{gathered}

Ømega

on function \begin{gathered}\\\tt \psi\end{gathered}

ψ

returns a constant \begin{gathered}\\\tt \omega\end{gathered}

ω

times the function itself.

Comparing each option one by one:

\dfrac{d( \log x)}{dx} = \dfrac{1}{x}

dx

d(logx)

=

x

1

# is not.

\begin{gathered}\\\tt \dfrac{d( e^x)}{dx} = 1 . e^x\end{gathered}

dx

d(e

x

)

=1.e

x

# is

\begin{gathered}\\\tt \dfrac{d \ sinx }{dx} =cos(x)\end{gathered}

dx

d sinx

=cos(x)

#is not

\begin{gathered}\\\tt \dfrac{d(xn)}{dx} =n\end{gathered}

dx

d(xn)

=n

# is not.