Math Problem Statement

If h(x) = f(g(x)), find h'(1) given the values of f(x), f'(x), g(x), and g'(x) at x = 1.

Solution

The problem asks to find h(1)h'(1) given that h(x)=f(g(x))h(x) = f(g(x)) and the table of values for f(x)f(x), f(x)f'(x), g(x)g(x), and g(x)g'(x).

Solution:

We can apply the chain rule to differentiate h(x)=f(g(x))h(x) = f(g(x)):

h(x)=f(g(x))g(x)h'(x) = f'(g(x)) \cdot g'(x)

Now, to find h(1)h'(1), we need the values of f(g(1))f'(g(1)) and g(1)g'(1) from the table.

From the table:

  • g(1)=1g(1) = 1
  • g(1)=2g'(1) = 2
  • f(1)=2f'(1) = 2 (because g(1)=1g(1) = 1)

Now substitute these into the chain rule formula:

h(1)=f(g(1))g(1)=f(1)g(1)=22=4h'(1) = f'(g(1)) \cdot g'(1) = f'(1) \cdot g'(1) = 2 \cdot 2 = 4

Thus, h(1)=4h'(1) = 4.


Would you like more details or have any other questions?

Here are 5 related questions you might find useful:

  1. What is the chain rule and how is it used in calculus?
  2. How would the derivative change if h(x)=f(x)g(x)h(x) = f(x) \cdot g(x) instead?
  3. What would happen if g(x)g'(x) were a constant function?
  4. Can you explain why we evaluated ff' at g(1)g(1) in this problem?
  5. How do we handle derivatives when functions are composed more than once (e.g., f(g(h(x)))f(g(h(x))))?

Tip: Always ensure you understand the chain rule, as it is essential for differentiating composite functions effectively.

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Composite Functions

Formulas

Chain rule: h'(x) = f'(g(x)) * g'(x)

Theorems

Chain rule for derivatives

Suitable Grade Level

Grades 11-12