The slope of the tangent to the graph of the function y = f (x) at the point (-2; 7) is 4. Find f ‘(- 2).

As you know, for a given function y = f (x), the equation of the tangent at the point x = x0 in general form is written as: y = f (x0) + f ꞌ (x0) * (x – x0). In the task, the slope of the tangent drawn to the graph of the function y = f (x) at the point (–2; 7) is specified, which is 4. It is required to find the value of the derivative f ꞌ (x) of the function f (x) for the value of the variable x = – 2.
Let’s expand the brackets in the above tangent equation. Then, we have: y = f ꞌ (x0) * x + f (x0) – f ꞌ (x0) * x0. Let us compare to this equation the equation of a straight line with a slope y = k * x + b, where the slope of a straight line is called a numerical coefficient k. Therefore, k = f ꞌ (x0) and b = f (x0) – f ꞌ (x0) * x0.
Thus, for our example, according to the conditions of the assignment, k = f ‘(–2) = 4. The very equation of the tangent drawn to the graph of the function y = f (x) at the point (–2; 7), which is equal to 4, has the form: y = 7 + 4 * (x – (–2)) or y = 4 * x + 15.
Answer: f ‘(–2) = 4.