Sure, let’s walk through the process of finding the derivative of the given function [tex]\( y = x^3 + x^2 + 1 \)[/tex].
1. Identify the function:
The function given is:
[tex]\[
y = x^3 + x^2 + 1
\][/tex]
2. Apply the power rule of differentiation:
The power rule states that if you have a term in the form [tex]\( x^n \)[/tex], its derivative is [tex]\( nx^{n-1} \)[/tex].
3. Differentiate each term individually:
- For the term [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{d}{dx}(x^3) = 3x^2
\][/tex]
- For the term [tex]\( x^2 \)[/tex]:
[tex]\[
\frac{d}{dx}(x^2) = 2x
\][/tex]
- For the constant term [tex]\( 1 \)[/tex]:
[tex]\[
\frac{d}{dx}(1) = 0
\][/tex]
4. Combine the results:
Adding the derivatives of each individual term, we get:
[tex]\[
\frac{d y}{d x} = 3x^2 + 2x
\][/tex]
So, the derivative of the function [tex]\( y = x^3 + x^2 + 1 \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
\frac{d y}{d x} = 3x^2 + 2x
\][/tex]
