Chain Rule

In Leibniz notation, if $y = f(u)$ and $u = g(x)$:

Concept Explanation

The Chain Rule describes how rates of change propagate through nested systems. If variable $y$ changes $A$ times faster than $u$, and $u$ changes $B$ times faster than $x$, then $y$ changes $A \times B$ times faster than $x$.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Used to compute rates of change in multi-component engineering systems, such as finding torque transmission ratios in mechanical gearboxes, modeling pressure changes in fluid systems, or predicting thermochemical reactions.

Example Problem

Differentiate $y = \sin(3x^2)$ with respect to $x$.

Solution

Let $u = 3x^2$ (inner function) and $y = \sin(u)$ (outer function).

Step 1: Find the derivative of the outer function with respect to $u$

$$\frac{dy}{du} = \cos(u)$$

Step 2: Find the derivative of the inner function with respect to $x$

$$\frac{du}{dx} = 6x$$

Step 3: Combine them using the Chain Rule

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(u) \cdot 6x$$

Step 4: Substitute back $u = 3x^2$

$$\frac{dy}{dx} = 6x \cos(3x^2)$$

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Connections & References

• Parent Concepts: Derivatives
• Sub-concepts: Implicit Differentiation
• Course Links: Calc-1 Session 08