Question

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.

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:

\(c = 6\)