Answer :
Certainly! Let's find the derivative of the function [tex]\( y = 4x^4 + 3x^3 + 1 \)[/tex].
### Step-by-Step Solution
1. Understand the Function:
The function given is [tex]\( y = 4x^4 + 3x^3 + 1 \)[/tex].
2. Apply Derivative Rules:
- When differentiating a polynomial, you use the power rule: [tex]\( \frac{d}{dx}[x^n] = nx^{n-1} \)[/tex].
- Differentiate term by term.
3. Differentiate Each Term:
- The first term [tex]\( 4x^4 \)[/tex]:
[tex]\[
\frac{d}{dx}[4x^4] = 4 \times 4x^{4-1} = 16x^3
\][/tex]
- The second term [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{d}{dx}[3x^3] = 3 \times 3x^{3-1} = 9x^2
\][/tex]
- The third term, a constant [tex]\( 1 \)[/tex]:
- The derivative of a constant is always 0.
[tex]\[
\frac{d}{dx}[1] = 0
\][/tex]
4. Combine the Derivatives:
- Combine the results from each term:
[tex]\[
\frac{dy}{dx} = 16x^3 + 9x^2 + 0
\][/tex]
5. Simplified Derivative:
- The derivative is:
[tex]\[
16x^3 + 9x^2
\][/tex]
Therefore, the correct derivative of the function [tex]\( y = 4x^4 + 3x^3 + 1 \)[/tex] is [tex]\( 16x^3 + 9x^2 \)[/tex].
Among the answer choices provided, [tex]\( 16x^3 + 9x^2 \)[/tex] matches the derivative calculation.
### Step-by-Step Solution
1. Understand the Function:
The function given is [tex]\( y = 4x^4 + 3x^3 + 1 \)[/tex].
2. Apply Derivative Rules:
- When differentiating a polynomial, you use the power rule: [tex]\( \frac{d}{dx}[x^n] = nx^{n-1} \)[/tex].
- Differentiate term by term.
3. Differentiate Each Term:
- The first term [tex]\( 4x^4 \)[/tex]:
[tex]\[
\frac{d}{dx}[4x^4] = 4 \times 4x^{4-1} = 16x^3
\][/tex]
- The second term [tex]\( 3x^3 \)[/tex]:
[tex]\[
\frac{d}{dx}[3x^3] = 3 \times 3x^{3-1} = 9x^2
\][/tex]
- The third term, a constant [tex]\( 1 \)[/tex]:
- The derivative of a constant is always 0.
[tex]\[
\frac{d}{dx}[1] = 0
\][/tex]
4. Combine the Derivatives:
- Combine the results from each term:
[tex]\[
\frac{dy}{dx} = 16x^3 + 9x^2 + 0
\][/tex]
5. Simplified Derivative:
- The derivative is:
[tex]\[
16x^3 + 9x^2
\][/tex]
Therefore, the correct derivative of the function [tex]\( y = 4x^4 + 3x^3 + 1 \)[/tex] is [tex]\( 16x^3 + 9x^2 \)[/tex].
Among the answer choices provided, [tex]\( 16x^3 + 9x^2 \)[/tex] matches the derivative calculation.
