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]
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]