Differentiation: The Chain Rule

The Chain Rule is used for differentiating composite functions $y = f(g(x))$ .
If $y = f(u)$ and $u = g(x)$ , then the derivative is given by the formula: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
Conceptual approach: Think of composite functions as having an "inside" and an "outside." The derivative is the derivative of the outer function (at the inner function $u$ ) multiplied by the derivative of the inner function.

A special case of the Chain Rule used for functions of the form $y = [u(x)]^n$ .
Rule: If $y = [u(x)]^n$ , then: $\frac{dy}{dx} = n[u(x)]^{n-1} \cdot \frac{du}{dx}$
Alternatively expressed as: $\frac{d}{dx} [u^n] = nu^{n-1} u'$
Example 2: To find the derivative of $y = (x^2 + 1)^3$ : $\frac{dy}{dx} = 3(x^2 + 1)^2 \cdot (2x) = 6x(x^2 + 1)^2$

Simplified forms are necessary for applications. Techniques include factoring out least powers and combining terms into single fractions.
Example 6 ( $y = x^2 \sqrt{1 - x^2}$ ): Derivative involves the Product Rule and General Power Rule, simplifying to: $y' = \frac{x(2 - 3x^2)}{\sqrt{1 - x^2}}$
Example 7 ( $f(x) = \left(\frac{3x - 1}{x^2 + 3}\right)^2$ ): Derivative uses the General Power Rule combined with the Quotient Rule, resulting in: $f'(x) = \frac{2(3x - 1)(-3x^2 + 2x + 9)}{(x^2 + 3)^3}$

Derivatives represent rates of change in real-life models.
Example 4: Finding the tangent line to $y = \sqrt[3]{(x^2 + 4)^2}$ at $x = 2$ . At this point, $y = 4$ .
Example 8 (Apple Performance): Sales per share $S$ for Apple from 2004 ( $t = 4$ ) through 2013 were analyzed using the General Power Rule to find the rate of change $\frac{dS}{dt}$ for years 2006, 2009, and 2012.

Constant Rule: $\frac{d}{dx} [c] = 0$
Constant Multiple Rule: $\frac{d}{dx} [cu] = c \frac{du}{dx}$
Sum and Difference Rules: $\frac{d}{dx} [u \pm v] = \frac{du}{dx} \pm \frac{dv}{dx}$
Product Rule: $\frac{d}{dx} [uv] = u \frac{dv}{dx} + v \frac{du}{dx}$
Quotient Rule: $\frac{d}{dx} \left[\frac{u}{v}\right] = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$
Standard Power Rule: $\frac{d}{dx} [x^n] = nx^{n-1}$
General Power Rule: $\frac{d}{dx} [u^n] = nu^{n-1} \frac{du}{dx}$