https://file.aiquickdraw.com/mathbot/file/1725941396399bnckwhw1.jpg

The problem asks to find the derivative of the given function \( y = 4x^3 + 3x - 1 \) and evaluate it at \( x = 2 \).

### Step 1: Find the derivative of \( y \)
We differentiate each term of the function \( y = 4x^3 + 3x - 1 \) with respect to \( x \):

\[
\frac{dy}{dx} = \frac{d}{dx}(4x^3) + \frac{d}{dx}(3x) + \frac{d}{dx}(-1)
\]

- The derivative of \( 4x^3 \) is \( 12x^2 \).
- The derivative of \( 3x \) is \( 3 \).
- The derivative of the constant \( -1 \) is \( 0 \).

So, the derivative is:

\[
\frac{dy}{dx} = 12x^2 + 3
\]

### Step 2: Evaluate the derivative at \( x = 2 \)

Now, substitute \( x = 2 \) into the derivative:

\[
\frac{dy}{dx} \bigg|_{x=2} = 12(2)^2 + 3 = 12(4) + 3 = 48 + 3 = 51
\]

Thus, the value of \( \frac{dy}{dx} \) at \( x = 2 \) is:

\[
\boxed{51}
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the second derivative of \( y = 4x^3 + 3x - 1 \)?
2. How do you compute the derivative of a polynomial function?
3. What is the derivative of \( y = 4x^3 + 3x - 1 \) at \( x = 1 \)?
4. What is the general form of the power rule in differentiation?
5. How does the derivative of a constant differ from that of a variable term?

### Tip:
Always remember that the derivative of a constant is zero, which can simplify calculations involving polynomials.

For y = 4x^3 + 3x - 1, find the following: dy/dx at x = 2.

Learn how to find the derivative of y = 4x^3 + 3x - 1 and evaluate it at x = 2. This problem covers key concepts in calculus, including the power rule and polynomial differentiation, suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation