Chain Rule (DP IB Analysis & Approaches (AA): HL): Revision Note

Chain rule

What is the chain rule?

• The chain rule states that if$\mathit{y}$ is a function of$\mathit{u}$ and$\mathit{u}$ is a function of$\mathit{x}$ 

  • i.e. if $y=g\left(u\right)$, where $u=f\left(x\right)$

  • then

    $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{u}}\mathbf{\times }\frac{\mathbf{d}\mathit{u}}{\mathbf{d}\mathit{x}}$
    

    • This is given in the formula booklet

• In function notation this could be written

$y=g\left(f\right(x\left)\right)$

$\frac{\mathrm{d}y}{\mathrm{d}x}=g^{\text{'}}\left(f\right(x\left)\right)f^{\text{'}}\left(x\right)$

How do I know when to use the chain rule?

•  Use the chain rule when you are trying to differentiate a composite function

  • i.e. a “function of a function”

• These can be identified as the variable (usually$x$) does not ‘appear alone’

  • $\sin x$ is not a composite function, because $x$ ‘appears alone’

  • $\sin (3x+2)$ is a composite function

    • $x$ is tripled and has 2 added to it before the sine function is applied

How do I use the chain rule?

• STEP 1
  Identify the two functions

  Rewrite $y$ as a function of$u$; $y=g\left(u\right)$

  Write $u$ as a function of$x$; $u=f\left(x\right)$
   

• STEP 2
  Differentiate $y$ with respect to $u$ to get$\frac{\mathrm{d}y}{\mathrm{d}u}$
  Differentiate $u$ with respect to $x$ to get$\frac{\mathrm{d}u}{\mathrm{d}x}$
   

• STEP 3
  Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}$ by applying the formula$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\times \frac{\mathrm{d}u}{\mathrm{d}x}$ and substituting $f\left(x\right)$ back in for $u$

Are there any standard results for using chain rule?

• There are five general results that can be useful