https://file.aiquickdraw.com/mathbot/file/1729774319689d796oc2d.jpg

The image discusses the **Chain Rule** for differentiation, which applies when differentiating composite functions of the form \( h(x) = f(g(x)) \). The Chain Rule states that:

\[
\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)
\]

### Breakdown of the Proof:
1. **Starting with the definition of the derivative**:
   \[
   \frac{d}{dx}f(g(x)) = \lim_{h \to 0} \frac{f(g(x+h)) - f(g(x))}{h}
   \]
   This is the standard difference quotient.

2. **Introducing the change in \( g(x) \)**:
   - By subtracting and adding \( g(x+h) - g(x) \) within the numerator, the expression is rewritten as:
   \[
   \lim_{h \to 0} \frac{f(g(x+h)) - f(g(x))}{h} = \lim_{h \to 0} \frac{f(g(x+h)) - f(g(x))}{g(x+h) - g(x)} \cdot \frac{g(x+h) - g(x)}{h}
   \]
   - This separates the change in \( g(x) \) from the change in \( f \), allowing the application of the **Chain Rule**.

3. **Applying the limits**:
   - Since \( g(x) \) is differentiable, \( \lim_{h \to 0} \frac{g(x+h) - g(x)}{h} = g'(x) \).
   - The limit \( \lim_{\Delta u \to 0} \frac{f(g(x) + \Delta u) - f(g(x))}{\Delta u} = f'(g(x)) \).

4. **Conclusion**:
   - Putting everything together results in:
   \[
   \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
   \]

Would you like more details or have any questions about this proof?

### 5 Related Questions:
1. How is the Chain Rule used in implicit differentiation?
2. What is the geometric interpretation of the Chain Rule?
3. How does the Chain Rule apply to higher-order derivatives?
4. Can you explain the difference between the Chain Rule and the Product Rule?
5. How can we apply the Chain Rule in cases where \( g(x) \) is a trigonometric or exponential function?

### Tip:
The Chain Rule can be extended to functions of multiple variables in multivariable calculus, which leads to the **multivariable Chain Rule**.

How can we prove the Chain Rule for differentiating composite functions?

This page explains the proof of the Chain Rule for differentiating composite functions. It covers key concepts such as the limit definition of the derivative, how the Chain Rule is applied, and why the rule works. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation