We use the product rule and the chain rule to find the derivative of y = e^4x sin 4x.
y = e^4x sin 4x
y' = (e^4x)(4sin 4x + cos 4x(4))
y' = 4e^4x(sin 4x + cos 4x)
Therefore, the first derivative of y=e^4x sin 4x is y' = 4e^4x(sin 4x + cos 4x).
Interpretation: The first derivative gives the instantaneous rate of change of the function at each point. In this case, the derivative represents the rate at which the amount of medicine remaining in a person's bloodstream changes with respect to time x. The function has a maximum or minimum when the first derivative is equal to zero.