Unit 3 Summary (College Board AP® Calculus AB): Revision Note

Differentiation of composite, implicit & inverse functions summary

Key facts & definitions

• $f^{\left(n\right)}\left(x\right)$ and $\frac{d^n}{d{x}^n}\left(f\left(x\right)\right)$ denote the nth derivative, which is the result when the function is differentiated $n$ times

• An equation is given implicitly if it is not written in the form $y=f\left(x\right)$

Key theorems

• The chain rule (for differentiating composite functions)

  • If $h\left(x\right)=f\left(g\left(x\right)\right)$

  • Then $h\text{'}\left(x\right)=f\text{'}\left(g\left(x\right)\right)·g\text{'}\left(x\right)$

• The inverse function theorem (for differentiating inverse functions)

  • If $g\left(x\right)=f^{-1}\left(x\right)$

  • Then $g\text{'}\left(x\right)=\frac{1}{f^{\text{'}}\left(g\left(x\right)\right)}$

  • This can be written as $\frac{dx}{dy}=\frac{1}{{\displaystyle \frac{dy}{dx}}}$

• The chain rule is used to differentiate terms in an implicit equation

  • $\frac{d}{dx}\left(f\left(y\right)\right)=f\text{'}\left(y\right)·\frac{dy}{dx}$

Key formulas

• Derivatives of inverse functions