Question

If \( y=(5x-2)e^x \), then \( \dfrac{d^2y}{dx^2} \) is equal to

• A. \( e^x(5x+8) \)
• B. \( e^x(5x-3) \)
• C. \( e^x(5x+5) \)
• D. \( e^x(5x+3) \)
• E. \( e^x(5x-5) \)

Hint:

For functions of the form \( (ax+b)e^x \), the derivative often stays in the form \( e^x(\text{linear expression}) \). Differentiate step by step and factor out \( e^x \) to simplify quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires finding the second derivative of a function. This is done by differentiating the function once to find the first derivative, and then differentiating the result again to find the second derivative. The function is a product of a polynomial and an exponential function, so the product rule for differentiation will be needed.
Step 2: Key Formula or Approach:
The product rule states that if \( y = u(x)v(x) \), then \( \frac{dy}{dx} = u'v + uv' \). We will apply this rule twice.
Step 3: Detailed Explanation:
First, find the first derivative, \( \frac{dy}{dx} \).
Let \( u = 5x - 2 \) and \( v = e^x \). Then \( u' = 5 \) and \( v' = e^x \).
\[ \frac{dy}{dx} = u'v + uv' = (5)(e^x) + (5x - 2)(e^x) \] Factor out \( e^x \):
\[ \frac{dy}{dx} = e^x(5 + 5x - 2) = e^x(5x + 3) \] Now, find the second derivative, \( \frac{d^2y}{dx^2} \), by differentiating \( \frac{dy}{dx} \).
Let \( u = e^x \) and \( v = 5x + 3 \). Then \( u' = e^x \) and \( v' = 5 \).
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( e^x(5x + 3) \right) = (e^x)(5x+3) + (e^x)(5) \] Factor out \( e^x \) again:
\[ \frac{d^2y}{dx^2} = e^x((5x+3) + 5) \] \[ \frac{d^2y}{dx^2} = e^x(5x + 8) \] Step 4: Final Answer:
The second derivative \( \frac{d^2y}{dx^2} \) is \( e^x(5x + 8) \). This corresponds to option (A).