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The problem in the image asks for the slope of the tangent line drawn to the graph of the function \( y = x^2 + 4x \) at the point \( A(1, 5) \).

To solve this:

1. **Find the derivative** of the function \( y = x^2 + 4x \), which gives the slope of the tangent line at any point on the curve.
   \[
   \frac{dy}{dx} = 2x + 4
   \]

2. **Evaluate the derivative at \( x = 1 \)** to find the slope of the tangent line at the point \( A(1, 5) \).
   \[
   \frac{dy}{dx} \Big|_{x=1} = 2(1) + 4 = 6
   \]

So, the slope of the tangent line at the point \( A(1, 5) \) is \( 6 \).

Would you like further clarification or details on any part of this solution? Here are some related questions you might consider:

1. How do you find the equation of the tangent line at \( A(1, 5) \)?
2. What is the geometric interpretation of the slope of a tangent line?
3. How do you calculate the slope of a tangent line for more complex functions?
4. Can you explain how to use implicit differentiation to find tangents in other cases?
5. How does the second derivative help determine concavity at a given point?

**Tip**: The derivative of a function at a point provides the slope of the tangent line to the curve at that point, giving insight into the curve's instantaneous rate of change.

y = x^2 + 4x funksiya grafiqqa A(1;5) nuqtada urinadigan to'g'ri chiziqning burchak koeffitsiyentini toping.

Learn how to find the slope of the tangent line to the graph of y = x^2 + 4x at the point A(1, 5). This problem involves basic calculus, including finding the derivative and evaluating it at a given point. Suitable for students in Grades 11-12.

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