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 $f(x)$.
$f(x)=\square$

Explanation:

Step1: Recall the product rule

The product rule states that if \(u(x)\) and \(v(x)\) are functions, then \((uv)^\prime = u^\prime v+uv^\prime\). Here \(u = e^{x}(3 + 5x)\) and \(v=(13 + 5x)\). First, find \(u^\prime\).
For \(u = e^{x}(3 + 5x)\), using the product rule again (\((e^{x})^\prime=e^{x}\) and \((3 + 5x)^\prime = 5\)), we have \(u^\prime=e^{x}(3 + 5x)+5e^{x}=e^{x}(8 + 5x)\)

Step2: Apply the product rule to \(f^\prime(x)=e^{x}(3 + 5x)(13 + 5x)\)

\(f^{\prime\prime}(x)=(e^{x}(8 + 5x))(13 + 5x)+e^{x}(3 + 5x)\times5\)

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Answer:

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