how to evaluate derivative of convolution integral? If the signal $x(t)$ has ordinary first derivative $\dot x(t)$, then $\dfrac{d}{dt}\big(x(t)\star y(t)\big)$ is:
  
  
  (a) $\dot x(t)y(t)$
  (b) $x(t)\dot y(t)$
  (c) $\dot x(t)\star y(t)$
  (d) $\dot x(t)\star \dot y(t)$
  


I solved it as follows: we know,

$$I(t)=x(t)\star y(t)=\displaystyle\int_{-\infty}^{\infty}x(\tau).y(t-\tau) d{\tau}\implies$$
$$\begin{align}\dfrac{dI(t)}{dt}&amp;=\dfrac{d}{dt}\displaystyle\int_{-\infty}^{\infty}x(\tau)y(t-\tau) d{\tau}\\&amp;=\displaystyle\int_{-\infty}^{\infty}\dfrac{\partial}{\partial t}x(\tau).y(t-\tau) d{\tau} \\&amp;=\displaystyle\int_{-\infty}^{\infty}x(\tau).\dfrac{\partial}{\partial t}y(t-\tau) d{\tau}=x(t)\star \dot y(t)\end{align}$$

So option $(b)$ should be correct but answer in the book is given as $(c)$ but I think both options $(b)$ and $(c)$ should be correct because in above
method I fixed $x(t)$ and shifted $y(t)$ but we can also do that other way
round fixing $y(t)$ and shifting $x(t)$,and we know by doing so,convolution
integral remain unaffected,so tell where i'm wrong ,any help would be appreciated
Note that option (b) is not correct, and that it is also not equal to what you came up with. Option (b) is just the multiplication of $x(t)$ and $y'(t)$, not the convolution. Your solution and option (c) are both correct, assuming that all derivatives exist and that the convolution integrals converge, because with that assumption the following holds:

$$\frac{d}{dt}\left(x(t)\star y(t)\right)=x'(t)\star y(t)=x(t)\star y'(t)\tag{1}$$

where $x'(t)$ and $y'(t)$ denote the derivatives of $x(t)$ and $y(t)$, respectively.

An ideal differentiator is just an LTI system, so it can be written as a convolution, and since convolution is associative, $(1)$ must hold.

This can also most easily be seen in the frequency domain. Differentiation corresponds to multiplication with $j\omega$, so we have

$$j\omega\left(X(\omega)Y(\omega)\right)=\left(j\omega X(\omega)\right)Y(\omega)=X(\omega)\left(j\omega Y(\omega)\right)\tag{2}$$

which is equivalent to $(1)$.