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To solve this, we need to find the derivative \(\frac{dy}{dx}\) of the function \(y = 2x^2 + 9x - 5\), and then evaluate the derivative at \(x = 8\).

### Step 1: Differentiate \(y = 2x^2 + 9x - 5\)
Using standard rules of differentiation:

- The derivative of \(2x^2\) is \(4x\) (since \(\frac{d}{dx}[x^n] = n x^{n-1}\)).
- The derivative of \(9x\) is 9.
- The derivative of the constant \(-5\) is 0.

Thus, the derivative is:

\[
\frac{dy}{dx} = 4x + 9
\]

### Step 2: Substitute \(x = 8\) into the derivative
Now, substitute \(x = 8\) into the expression for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = 4(8) + 9 = 32 + 9 = 41
\]

### Final Answer:
The value of \(\frac{dy}{dx}\) when \(x = 8\) is **41**.

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Would you like further clarification or more details? Here are some related questions for further exploration:

1. What is the geometric meaning of the derivative at a specific point?
2. How do you apply the power rule for differentiation in general?
3. Can we find the second derivative of \(y = 2x^2 + 9x - 5\)?
4. What are the differences between differentiation rules for polynomials and trigonometric functions?
5. How can we use the derivative to find the equation of the tangent line at \(x = 8\)?

**Tip:** Differentiation finds the slope of the tangent to the curve at a particular point, helping you analyze the rate of change at any given \(x\).

Find dy/dx for y = 2x^2 + 9x - 5 when x = 8.

Learn how to differentiate the polynomial function y = 2x^2 + 9x - 5 and find the value of the derivative when x = 8. This problem involves basic differentiation rules and the power rule, suitable for students in high school calculus.

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