Question

find the first and second derivatives of the given function.
$f(x)=e^{x}(3 + 5x)^{2}$
calculate $f(x)$.
$f(x)=e^{x}(3 + 5x)(13 + 5x)$
calculate $f(x)$.
$f(x)=$

Explanation:

Step1: Recall the product rule

The product rule states that if \(u(x)\) and \(v(x)\) are functions, then \((uv)^\prime=u^\prime v + uv^\prime\). Let \(u = e^{x}(3 + 5x)\) and \(v=(13 + 5x)\). First, find \(u^\prime\):
\(u^\prime=(e^{x}(3 + 5x))^\prime=e^{x}(3 + 5x)+5e^{x}=e^{x}(8 + 5x)\)
\(v^\prime = 5\)

Step2: Apply the product rule to \(f^\prime(x)=u\times v\)

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Answer:

\(f^{\prime\prime}(x)=e^{x}(5x + 7)(5x + 17)\)