To determine the derivative of the function $f(x) = 2x^2 - 5x + 11$ at a specific point $a$, we first need to find the general derivative $f'(x)$ of the function.

Step 1: Differentiate the function

We will differentiate $f(x) = 2x^2 - 5x + 11$ term by term.

• The derivative of $2x^2$ is $4x$ (using the power rule).
• The derivative of $-5x$ is $-5$ (the derivative of $x$ is 1, and the constant stays).
• The derivative of $11$ is $0$ (since it's a constant).

So, the derivative is:

$f'(x) = 4x - 5$

Step 2: Evaluate the derivative at $x = a$

To find the derivative at the specific point $x = a$, we substitute $a$ into the derivative function:

$f'(a) = 4a - 5$

Thus, the derivative of $f(x)$ at the point $x = a$ is $f'(a) = 4a - 5$.

If you'd like further clarification or have additional questions, feel free to ask!

Follow-up Questions:

1. What is the meaning of a derivative at a point on a graph?
2. How do you find the derivative of a more complex polynomial function?
3. Can the derivative be interpreted as a slope? How does that relate to the function's graph?
4. What does the second derivative tell you about a function's graph?
5. How would you apply this derivative to find the tangent line to the graph at a given point?

Tip: When differentiating polynomials, remember that the derivative of $ax^n$ is $nax^{n-1}$.