Advanced Placement Calculus AB covering limits, derivatives, and integrals.
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.
Multiply the derivative of the outer function by the derivative of the inner function: \[ \text{If } y = f(g(x)),\ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
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.
The chain rule and implicit differentiation are used in physics, economics, and engineering, wherever variables depend on each other in complex ways.
\[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]
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 \( \frac{dy}{dx} = -\frac{x}{y} \).
The chain rule handles nested functions; implicit differentiation finds derivatives when variables are mixed.