502ChainRule

The Chain Rule

The chain rule states that: math \dfrac {dy}{dx} = \dfrac {dy}{du} \times \dfrac {du}{dx} math

Notice that if you treat these as fractions, the right side simplifies to give the left side

Example Differentiate math y = \Big( 2x^3 - 3x \Big) ^4 \qquad \big[ \text{ Equation 1} \big] math

math \text {Let } \; u = 2x^3 - 3x math

Differentiate u with respect to x math \dfrac {du}{dx} = 6x^2 - 3 \qquad \quad \big[ \text{Equation 2} \big] math

Equation **1** becomes math y=u^4 math

Differentiate y with respect to u math \dfrac {dy}{du} = 4u^3 \qquad \qquad \qquad \big[ \text{Equation 3} \big] math

Replace u in equation **3**. math \dfrac {dy}{du} = 4 \Big( 2x^3-3x \Big)^3 \quad \; \big[ \text{ Equation 4} \big] math

Now put **4** and **2** together into the Chain Rule math \dfrac {dy}{dx} = \dfrac {dy}{du} \times \dfrac {du}{dx} math

math \dfrac {dy}{dx} = 4 \left( 2x^3 -3x \right)^3 \left( 6x^2 - 3 \right) math

Your calculator can do questions like this

**See also** [|www.mathsonline.com.au] Y12Advanced --> Calculus --> Introductory Calculus --> Lesson 4