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.