# Calculus Cheat Sheet.pdf - Common Derivatives =1 = Limits...

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Limits & Derivatives Cheat Sheet Properties of Limits lim 𝑥→? ?? 𝑥 = ? lim 𝑥→? ?(𝑥) lim 𝑥→? [? 𝑥 ± ? 𝑥 ] = lim 𝑥→? ?(𝑥) ± lim 𝑥→? ?(𝑥) lim 𝑥→? [? 𝑥 ? 𝑥 ] = lim 𝑥→? ?(𝑥) lim 𝑥→? ?(𝑥) lim 𝑥→? ? 𝑥 ? 𝑥 = lim 𝑥→? ?(𝑥) lim 𝑥→? ?(𝑥) 𝑖? lim 𝑥→? ?(𝑥) ≠ 0 lim 𝑥→? [? 𝑥 ] ? = lim 𝑥→? ? 𝑥 ? Limit Evaluations At ± lim 𝑥→+∞ ? 𝑥 = lim 𝑥→−∞ ? 𝑥 = 0 lim 𝑥→∞ ln 𝑥 = ∞ and lim 𝑥→0 + ln 𝑥 = −∞ if r > 0: lim 𝑥→∞ ? 𝑥 𝑟 = 0 if r > 0 & ∀𝑥 > 0 𝑥 𝑟 ∈ ℝ lim 𝑥→−∞ ? 𝑥 𝑟 = 0 lim 𝑥→±∞ 𝑥 𝑟 = ∞ for even r lim 𝑥→+∞ 𝑥 𝑟 = ∞ and lim 𝑥→−∞ 𝑥 𝑟 = −∞ for odd r L’Hopital’s Rule If lim 𝑥→? ?(𝑥) ?(𝑥) = 0 0 or ±∞ ±∞ then lim 𝑥→? ?(𝑥) ?(𝑥) = lim 𝑥→? ?′(𝑥) ?′(𝑥) Common Derivatives ? ?𝑥 𝑥 = 1 ? ?𝑥 [?? 𝑥 ] = ? ? ?𝑥 [? 𝑥 ] ? ?𝑥 ?𝑥 = ? ? ?𝑥 ?𝑥 ? = ??𝑥 ?−1 ? ?𝑥 ? = 0 ? ?𝑥 [? 𝑥 ] ? = ?[? 𝑥 ] ?−1 ?′(𝑥) ? ?𝑥 1 𝑥 ? = −?𝑥 ?+1 = − ? 𝑥 ?+1 Derivative Definition ? ?𝑥 ? 𝑥 = ? 𝑥 = lim ℎ→0 ? 𝑥 + ℎ − ?(𝑥) Product Rule ? 𝑥 ? 𝑥 = ? 𝑥 ? 𝑥 + ? 𝑥 ?′(𝑥) Quotient Rule ? ?𝑥 ? 𝑥 ? 𝑥 = ? 𝑥 ? 𝑥 − ? 𝑥 ?′(𝑥) [? 𝑥 ] 2 Chain Rule ? ?𝑥 ? ? 𝑥 = ? ? 𝑥 ?′(𝑥) Derivatives of Trigonometric Functions ? ?𝑥 sin 𝑥 = cos 𝑥 ? ?𝑥 sec 𝑥 = sec 𝑥 tan 𝑥 ? ?𝑥 cos 𝑥 = − sin𝑥 ? ?𝑥 csc 𝑥 = − csc 𝑥 cot 𝑥 ? ?𝑥 tan 𝑥 = sec 2 𝑥 ? ?𝑥 cot 𝑥 = −csc 2 𝑥 Derivatives of Exponential & Logarithmic Functions ? ?𝑥 ? 𝑥 = 𝑥 ? ?𝑥 ? 𝑥 = ? 𝑥 ln ? ? ?𝑥 ln |𝑥| = 1 𝑥 ? ?𝑥 ln 𝑥 = 1 𝑥 , 𝑥 > 0 ? ?𝑥 log ? 𝑥 = 1 𝑥 ln ? ?