The image shows part of a math problem that involves finding the derivative of a function. Specifically, the problem is:

$f(x) = x^2 + 7$

The problem is asking to solve using derivatives, possibly to find the slope at a given point, or to find the derivative function itself.

Steps to Solve Using Derivatives:

1. Find the Derivative of the Function:

   • The function given is $f(x) = x^2 + 7$.
   • To find the derivative, $f'(x)$, apply the power rule to each term.
   • The derivative of $x^2$ is $2x$.
   • The derivative of the constant $7$ is $0$.

   Therefore, the derivative of the function is: $f'(x) = 2x$

2. Evaluate the Derivative at a Specific Point (if required):

   • If you need to find the slope at a particular point $x = a$, substitute $a$ into the derivative function.
   • For example, if $x = 2$, then: $f'(2) = 2(2) = 4$ This means the slope of the function at $x = 2$ is 4.

If the problem asks for finding the slope at a particular point, you can substitute that value of $x$ into the derivative. Otherwise, the derivative function $f'(x) = 2x$ provides the slope of the tangent line at any point $x$ on the curve.

Would you like more details on this solution, or do you have any specific questions?

Follow-up Questions:

1. What does the derivative $f'(x) = 2x$ represent in a graph of $f(x) = x^2 + 7$?
2. How does the constant term in $f(x)$ affect the derivative?
3. Can you find the second derivative of $f(x) = x^2 + 7$?
4. What is the significance of a derivative being zero at a point on the curve?
5. How would you use derivatives to determine if the function $f(x) = x^2 + 7$ has any local maxima or minima?

Tip:

When differentiating, always start by applying basic rules like the power rule and sum rule, and then evaluate the derivative at specific points to find slopes of tangent lines if needed.