the derivative of ( f(x) = 3e^{2x + 1} ) is ( f(x) = ce^{2x + 1} ). find the value of c. give an exact…

the derivative of ( f(x) = 3e^{2x + 1} ) is ( f(x) = ce^{2x + 1} ). find the value of c. give an exact answer as an integer.

the derivative of ( f(x) = 3e^{2x + 1} ) is ( f(x) = ce^{2x + 1} ). find the value of c. give an exact answer as an integer.

Answer

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = e^{u}), then (y^\prime=e^{u}\cdot u^\prime). For (f(x)=3e^{2x + 1}), let (u = 2x+1). Then (u^\prime=2). [ f^\prime(x)=3\cdot e^{2x + 1}\cdot(2x + 1)^\prime ]

Step2: Calculate the derivative

Since ((2x + 1)^\prime=2), we substitute this into the formula. [ f^\prime(x)=3\cdot e^{2x+1}\cdot2=6e^{2x + 1} ]

Answer:

(c = 6)