https://file.aiquickdraw.com/mathbot/file/1725909044243qfs023w0.jpg

how does do we get the slope of f' at (x_0, f(x_0))

The slope of the function \( f \) at the point \( (x_0, f(x_0)) \) is determined by the derivative \( f'(x_0) \). This is because the derivative \( f'(x_0) \) represents the instantaneous rate of change of the function at \( x_0 \), which is the same as the slope of the tangent line to the curve at that point.

### Step-by-step derivation of the slope:

1. **Point-Slope Formula**: 
   The general form of a line's equation using the point-slope formula is:
   \[
   y = y_0 + m(x - x_0)
   \]
   where \( (x_0, y_0) \) is a point on the line and \( m \) is the slope.

2. **Tangent Line at \( (x_0, f(x_0)) \)**:
   At the point \( (x_0, f(x_0)) \), the slope of the tangent line is given by the derivative of \( f \) at \( x_0 \), denoted \( f'(x_0) \). 

3. **Substitute into the Point-Slope Formula**:
   Plugging \( m = f'(x_0) \) and \( y_0 = f(x_0) \) into the point-slope formula, we get:
   \[
   y = f(x_0) + f'(x_0)(x - x_0)
   \]
   This is the equation of the tangent line at the point \( (x_0, f(x_0)) \).

4. **Resulting Equation**:
   The equation becomes:
   \[
   y = f(x_0) + f'(x_0)(x - x_0) = L(x)_{x_0}
   \]
   where \( L(x) \) represents the linear approximation or the tangent line at \( x_0 \).

Thus, the slope \( f'(x_0) \) at the point \( (x_0, f(x_0)) \) is derived from the fact that the derivative of the function at \( x_0 \) gives the rate of change (or slope) of the function at that point.

---

Would you like further details or have any questions on this explanation?

Here are 5 relative questions to deepen your understanding:
1. How is the derivative related to the concept of a tangent line?
2. What is the geometric interpretation of \( f'(x_0) \)?
3. How do we compute the derivative \( f'(x) \) from first principles?
4. Why is the point-slope form used to derive the equation of the tangent line?
5. Can the equation of a secant line between two points help in understanding the concept of a tangent line?

**Tip:** When interpreting derivatives, always think of them as the slope of the tangent to the curve at any specific point.

how do we get the slope of f' at (x_0, f(x_0))?

Explore how to find the slope of the derivative of a function at a specific point. This problem covers key calculus concepts such as derivatives, tangent lines, and the point-slope formula. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation