AP Calculus AB
Advanced Placement Calculus AB covering limits, derivatives, and integrals.
The Chain Rule and Implicit Differentiation
What Is the Chain Rule?
The chain rule allows us to differentiate composite functions—functions inside of other functions. If you have \( f(g(x)) \), the chain rule helps you find its derivative.
How Does It Work?
Multiply the derivative of the outer function by the derivative of the inner function: \[ \text{If } y = f(g(x)),\ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
Implicit Differentiation
Sometimes, functions are not written explicitly as \( y = f(x) \). Implicit differentiation allows us to find derivatives even when \( y \) and \( x \) are tangled together in an equation.
Steps for Implicit Differentiation
- Differentiate both sides of the equation with respect to \( x \).
- Remember to use the chain rule when differentiating terms with \( y \).
- Solve for \( \frac{dy}{dx} \).
Applications
The chain rule and implicit differentiation are used in physics, economics, and engineering, wherever variables depend on each other in complex ways.
Key Formula
\[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]
Examples
To differentiate \( y = (3x+2)^5 \), use the chain rule: \( 5(3x+2)^4 \cdot 3 \).
For the circle equation \( x^2 + y^2 = 25 \), implicit differentiation gives \( \frac{dy}{dx} = -\frac{x}{y} \).
In a Nutshell
The chain rule handles nested functions; implicit differentiation finds derivatives when variables are mixed.