#Derivatives of Composite Functions

#Table of Contents

1. Understanding Composite Functions
2. Using the Chain Rule
3. Worked Examples
4. Advanced Applications of the Chain Rule
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

#Understanding Composite Functions

When differentiating composite functions, we use the chain rule. Composite functions are of the form $f(g(x))$.

#The Chain Rule

If $y = f(u)$ and $u = g(x)$, then: $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$

In function notation, if $h(x) = f(g(x))$, then: $$h'(x) = f'(g(x)) \cdot g'(x)$$

The chain rule can be extended to longer chains of functions: $$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dt} \times \frac{dt}{dr} \times \frac{dr}{dx}$$

#Using the Chain Rule

#Example: Differentiating $y = (2x^3 + 4x)^6$

1. Identify the inside function: $u = 2x^3 + 4x$
2. Rewrite the original function: $y = u^6$
3. Differentiate the inside function with respect to $x$: $$\frac{du}{dx} = 6x^2 + 4$$
4. **Differentiate $y$ wi...