Welcome back to Professor Baker's Math Class! In Section 3-4, we tackle one of the most powerful and frequently used tools in Calculus: The Chain Rule. Up until now, we've been differentiating standard functions like $x^2$ or $\sin(x)$. But what happens when functions are stuck inside other functions, like $\sqrt{x^2+1}$ or $e^{\sin x}$? That is where the Chain Rule shines.

Understanding Composite Functions

The Chain Rule is designed for composite functions, which we denote as $F(x) = f(g(x))$. Think of these as having an "outer" function ($f$) and an "inner" function ($g$).

The rule states that to find the derivative $F'(x)$, you take the derivative of the outer function (leaving the inside alone) and multiply it by the derivative of the inner function.

The Formula:

$$ F'(x) = f'(g(x)) \cdot g'(x) $$

In Leibniz notation, if we let $u = g(x)$, the rule looks like a chain of fractions:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

Key Applications from Class Notes

1. The General Power Rule

When you have a function raised to a power, like $y = (x^3 - 1)^{100}$, you treat the parenthesis as your "inner" function ($u$).

  • Step 1: Bring the power down and subtract 1 from the exponent (Power Rule on the outside).
  • Step 2: Multiply by the derivative of the inside.
$$ \frac{d}{dx}[u^n] = n \cdot u^{n-1} \cdot u' $$

2. Trigonometric Functions

It is crucial to recognize the difference in notation, as seen in the notes:

  • $y = \sin(x^2)$: Here, the angle is squared. The outer function is sine, the inner is $x^2$.
  • $y = \sin^2(x)$: This means $(\sin x)^2$. The outer function is the power function, the inner is sine.

3. Exponential Functions

We extended our derivative rules to include exponentials combined with the chain rule. For the natural base $e$, the rule is straightforward:

$$ \frac{d}{dx}[e^u] = e^u \cdot u' $$

However, for any other base $b$ (like $2^x$ or $5^{x^2}$), we must include a natural log adjustment:

$$ \frac{d}{dx}[b^u] = b^u \cdot \ln(b) \cdot u' $$

Nested Chains

Sometimes, functions are nested three layers deep! As shown in the example $f(x) = \sin(\cos(\tan x))$, we simply apply the chain rule sequentially, working from the outermost shell to the innermost core:

$$ f'(x) = \cos(\cos(\tan x)) \cdot [-\sin(\tan x)] \cdot \sec^2(x) $$

Tips for Success

When working through the homework, identify your $u$ (the inner function) first. Whether you are dealing with radicals, exponents, or trig functions, the process remains the same: derivative of the outside $\times$ derivative of the inside. Keep practicing, and you will master these compositions in no time!