f(x)= x^2 + 10 can you find the rate of change

To find the rate of change of the function \( f(x) = x^2 + 10 \), we need to calculate its **derivative**. The derivative represents the rate of change of the function with respect to \( x \).

### Step-by-step:
Given:  
\[ f(x) = x^2 + 10 \]

To find the derivative, \( f'(x) \), we apply the power rule:
1. The derivative of \( x^2 \) is \( 2x \).
2. The derivative of a constant (in this case, 10) is 0.

So, the derivative is:
\[
f'(x) = 2x
\]

This means the rate of change of the function \( f(x) \) at any point \( x \) is \( 2x \).

### Conclusion:
The rate of change of \( f(x) = x^2 + 10 \) is given by \( f'(x) = 2x \). At any specific value of \( x \), you can substitute it into \( 2x \) to find the exact rate of change at that point.

Would you like to calculate the rate of change at a specific point or explore this further?

### Relative questions:
1. What is the rate of change of \( f(x) = x^2 + 10 \) at \( x = 3 \)?
2. How do you find the rate of change of a function involving a higher degree polynomial?
3. What is the difference between average rate of change and instantaneous rate of change?
4. How does the rate of change vary for \( f(x) = x^2 + 10 \) as \( x \) increases?
5. Can the rate of change of a quadratic function ever be constant?

### Tip:
The derivative tells you the **instantaneous rate of change** at any point. For quadratic functions, the rate of change increases or decreases linearly as \( x \) changes.

Explore how to find the rate of change of the quadratic function f(x) = x^2 + 10 by calculating its derivative. This step-by-step guide demonstrates the application of the power rule and explains the concepts of derivatives and rate of change, suitable for students in Grades 10-12.

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