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Select the correct answer.

If [tex]y = x^5 + 3x^2 + 7x[/tex], find [tex]\frac{d^2 y}{d x^2}[/tex].

A. [tex]60 x^2[/tex]

B. [tex]20 x^3 + 6[/tex]

C. [tex]5 x^4 + 6 x[/tex]

D. [tex]5 x^4 + 6 x + 7 x[/tex]

Answer :

To find the second derivative of the function [tex]\( y = x^5 + 3x^2 + 7x \)[/tex] with respect to [tex]\( x \)[/tex], we need to go through two differentiation steps.

Step 1: Find the First Derivative [tex]\(\frac{dy}{dx}\)[/tex]

We start by differentiating [tex]\( y = x^5 + 3x^2 + 7x \)[/tex] with respect to [tex]\( x \)[/tex].

- The derivative of [tex]\( x^5 \)[/tex] is [tex]\( 5x^4 \)[/tex].
- The derivative of [tex]\( 3x^2 \)[/tex] is [tex]\( 6x \)[/tex].
- The derivative of [tex]\( 7x \)[/tex] is [tex]\( 7 \)[/tex].

So, the first derivative is:
[tex]\[
\frac{dy}{dx} = 5x^4 + 6x + 7
\][/tex]

Step 2: Find the Second Derivative [tex]\(\frac{d^2y}{dx^2}\)[/tex]

Now, we differentiate [tex]\(\frac{dy}{dx} = 5x^4 + 6x + 7\)[/tex] with respect to [tex]\( x \)[/tex].

- The derivative of [tex]\( 5x^4 \)[/tex] is [tex]\( 20x^3 \)[/tex].
- The derivative of [tex]\( 6x \)[/tex] is [tex]\( 6 \)[/tex].
- The derivative of [tex]\( 7 \)[/tex] (a constant) is [tex]\( 0 \)[/tex].

So, the second derivative is:
[tex]\[
\frac{d^2y}{dx^2} = 20x^3 + 6
\][/tex]

Comparing this result with the given options, the correct answer is:
B. [tex]\(20x^3 + 6\)[/tex]

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