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: Write the given function clearly.
We are given \[ y=(5x-2)e^x \] We have to find the second derivative \[ \frac{d^2y}{dx^2} \] Since \( y \) is a product of two functions, we use the product rule.

Step 2: Recall the product rule.
If \[ y=u\cdot v \] then \[ \frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx} \] Here, \[ u=5x-2,\qquad v=e^x \] So, \[ \frac{du}{dx}=5,\qquad \frac{dv}{dx}=e^x \]

Step 3: Find the first derivative.
Applying the product rule, \[ \frac{dy}{dx} = (5x-2)e^x+e^x(5) \] \[ = e^x\bigl((5x-2)+5\bigr) \] \[ = e^x(5x+3) \]

Step 4: Differentiate again to find the second derivative.
Now, \[ \frac{dy}{dx}=e^x(5x+3) \] Again this is a product of two functions, so we apply the product rule once more.
Let \[ u=e^x,\qquad v=5x+3 \] Then \[ \frac{du}{dx}=e^x,\qquad \frac{dv}{dx}=5 \]

Step 5: Apply the product rule to \( e^x(5x+3) \).
Thus, \[ \frac{d^2y}{dx^2} = e^x(5x+3)+e^x(5) \] \[ = e^x\bigl((5x+3)+5\bigr) \] \[ = e^x(5x+8) \]

Step 6: Verify the simplification carefully.
From the previous step, \[ (5x+3)+5=5x+8 \] So the second derivative is correctly simplified as \[ \frac{d^2y}{dx^2}=e^x(5x+8) \]

Step 7: Final conclusion.
Therefore, \[ \boxed{\frac{d^2y}{dx^2}=e^x(5x+8)} \] Hence, the correct option is \[ \boxed{(1)\ e^x(5x+8)} \]