AP Calculus AB · Summary

Unit 3 summary

1. Chain rule: $\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)$. Keep the inside, multiply by its derivative, and read the outer derivative at $g(x)$.
2. $\frac{d}{dx}e^{u} = e^{u}u'$, $\frac{d}{dx}\ln u = \frac{u'}{u}$, $\frac{d}{dx}b^x = b^x\ln b$.
3. Implicit differentiation: differentiate both sides, attach $\frac{dy}{dx}$ to every $y$ term, use the product rule on terms like $xy$, then solve for $\frac{dy}{dx}$.
4. With $\frac{dy}{dx} = \frac{N}{D}$: horizontal tangent where $N = 0$ and $D \ne 0$; vertical tangent where $D = 0$ and $N \ne 0$.
5. Inverse: if $f(a) = b$, then $(f^{-1})'(b) = \frac{1}{f'(a)}$. Find $a$ from the outputs, not by plugging $b$ into $f'$.
6. $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}$, $\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1 - x^2}}$, $\frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$.
7. A second derivative is the derivative of the derivative; found implicitly, substitute $\frac{dy}{dx}$ back in.