Since `W_0=10^(-12)\ "W/m"^2`, and using the logarithm law for the log of a fraction, we can write:

`P=10\ log\ (W)/(10^-12)`

`=10(log\ W-log\ 10^(-12))`

Now, using the formula for the derivative of a logarithm, and because log 10^{-12} is a constant, we have:

`(dP)/(dt)=10([1/W\ log_10e](dW)/(dt)-0)`

`=10[1/W\ log_10e](dW)/(dt)`

Now we substitute our given values for *W* and `(dW)/dt` from the question:

`(dP)/(dt)=10([1/7.2\ log_10e](0.5))`

`=0.302\ "dB"//"s"`

The units are dB/s since the sound pressure (in dB) is changing over time.

Please support IntMath!