https://file.aiquickdraw.com/mathbot/file/1728527019835wg33rk0u.jpg

The problem presented asks you to work with two differentiable functions \( f \) and \( g \), where \( h(x) = f(g(x)) \), which means \( h(x) \) is a composition of the functions \( f \) and \( g \). You are also given an equation for the tangent line to the graph of \( g \) at a specific point and asked to find both the value and derivative of \( h \) at \( x = 4 \).

### Let's break it down:

1. **Given Data:**
   - \( h(x) = f(g(x)) \)
   - The equation of the tangent to the graph of \( g \) at the point \( (4, 6) \) is \( y = 4x - 10 \).
     - This tells us that the slope of the tangent line is \( 4 \), meaning \( g'(4) = 4 \).
     - Also, since the tangent line passes through \( (4, 6) \), we know \( g(4) = 6 \).
   - The equation of the tangent line to the graph of \( f \) at the point \( (6, 2) \) is \( y = -3x + 20 \).
     - The slope of this line is \( -3 \), meaning \( f'(6) = -3 \).
     - Also, since the tangent line passes through \( (6, 2) \), we know \( f(6) = 2 \).

### Part (a): Calculate \( h(4) \) and \( h'(4) \).

- **For \( h(4) \):**
  \[
  h(4) = f(g(4)) = f(6)
  \]
  From the given information, \( f(6) = 2 \). Therefore:
  \[
  h(4) = 2
  \]

- **For \( h'(4) \):**
  Using the chain rule:
  \[
  h'(x) = f'(g(x)) \cdot g'(x)
  \]
  So at \( x = 4 \):
  \[
  h'(4) = f'(g(4)) \cdot g'(4) = f'(6) \cdot g'(4)
  \]
  From the given information, \( f'(6) = -3 \) and \( g'(4) = 4 \). Therefore:
  \[
  h'(4) = (-3) \cdot 4 = -12
  \]

### Final answers:
- \( h(4) = 2 \)
- \( h'(4) = -12 \)

Do you need further clarifications or details?

Here are five related questions to deepen your understanding:
1. How is the chain rule applied to composite functions like \( h(x) = f(g(x)) \)?
2. Why does the slope of the tangent line correspond to the derivative of a function at a specific point?
3. What information can we derive from the equation of a tangent line to a curve?
4. How can we interpret the derivative of a composition of functions geometrically?
5. What is the significance of the values of \( f'(x) \) and \( g'(x) \) in the context of the chain rule?

**Tip:** When differentiating a composite function, always remember to multiply the outer derivative by the derivative of the inner function.

Assume f and g are differentiable functions with h(x) = f(g(x)). Suppose the equation of the line tangent to the graph of g at the point (4,6) is y = 4x - 10 and the equation of the line tangent to the graph of f at (6,2) is y = -3x + 20.

Part a: Calculate h(4) and h'(4).
Part b: Determine an equation of the line tangent to the graph of h at the point on the graph where x = 4.

Learn how to calculate the derivative of composite functions using the chain rule. This problem involves finding the derivative and equation of the tangent line for a given function at a specific point. Suitable for students in Grades 11-12, this problem reinforces concepts in differentiation and the application of the chain rule.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation