AP Calculus AB · Summary

Unit 2 summary

  1. f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}: the slope of the tangent line and the instantaneous rate of change. A limit in this form is a derivative in disguise.
  2. From a table, estimate a derivative with the closest inputs on each side of the point.
  3. Differentiable at a point implies continuous there, not the reverse. No derivative at a corner, a cusp, a vertical tangent or a discontinuity.
  4. Power rule ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1} (rewrite roots and reciprocals first); ddxex=ex\frac{d}{dx}e^x = e^x; ddxln⁡x=1x\frac{d}{dx}\ln x = \frac{1}{x}.
  5. Product rule f′g+fg′f'g + fg'; quotient rule f′g−fg′g2\frac{f'g - fg'}{g^2}. Never multiply or divide the separate derivatives.
  6. sin⁡→cos⁡\sin \to \cos, cos⁡→−sin⁡\cos \to -\sin, tan⁡→sec⁡2\tan \to \sec^2, sec⁡→sec⁡tan⁡\sec \to \sec\tan; the co- functions take a minus sign. Radians only.
  7. Tangent line at aa: y=f(a)+f′(a)(x−a)y = f(a) + f'(a)(x - a).
Practice AP Calculus AB questions