Derivatives-Formula-Sheet.pdf

Calculus 1 – Derivatives Formula Sheet: Basic Derivatives: 𝑑 𝑑 𝑑 [𝒄] = 0 [𝒙] = 1 [𝒄𝒙] = 𝑐 𝑑𝑥 𝑑𝑥 𝑑𝑥...

Calculus 1 – Derivatives Formula Sheet: Basic Derivatives: 𝑑 𝑑 𝑑 [𝒄] = 0 [𝒙] = 1 [𝒄𝒙] = 𝑐 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑 [𝒄 ∗ 𝒇(𝒙)] = 𝑐 ∗ 𝑓′(𝑥) 𝑑𝑥 Trigonometric Derivatives: 𝑑 𝑑 [𝐬𝐢𝐧 𝒙] = cos 𝑥 [𝐜𝐨𝐬 𝒙] = − sin 𝑥 𝑑𝑥 𝑑𝑥 𝑑 𝑑 [𝐭𝐚𝐧 𝒙] = 𝑠𝑒𝑐 2 𝑥 [𝐜𝐨𝐭 𝒙] = −𝑐𝑠𝑐 2 𝑥 𝑑𝑥 𝑑𝑥 𝑑 𝑑 [𝐬𝐞𝐜 𝒙] = sec 𝑥 tan 𝑥 [𝐜𝐬𝐜 𝒙] = − csc 𝑥 cot 𝑥 𝑑𝑥 𝑑𝑥 The Power Rule: 𝑑 𝒏 [𝒙 ] = 𝑛𝑥 𝑛−1 𝑑𝑥 The Product Rule: 𝑑 [𝒖𝒗] = 𝑢′ 𝑣 + 𝑢𝑣′ 𝑑𝑥 𝑑 [𝒖𝒗𝒘] = 𝑢′ 𝑣𝑤 + 𝑢𝑣 ′ 𝑤 + 𝑢𝑣𝑤′ 𝑑𝑥 The Quotient Rule: 𝑑 𝒖 𝑣𝑢′ − 𝑢𝑣′ [ ]= 𝑑𝑥 𝒗 𝑣2 The Reciprocal Rule: 𝑑 𝟏 −𝑢′ [ ]= 2 𝑑𝑥 𝒖 𝑢 www.Video-Tutor.net The Chain Rule: 𝒅𝒚 𝑑𝑦 𝑑𝑢 = ∗ 𝒅𝒙 𝑑𝑢 𝑑𝑥 𝑑 [𝒇(𝒈(𝒙))] = 𝑓 ′ (𝑔(𝑥)) ∗ 𝑔′(𝑥) 𝑑𝑥 𝑑 [𝒇(𝒈(𝒖))] = 𝑓 ′ (𝑔(𝑢)) ∗ 𝑔′ (𝑢) ∗ 𝑢′ 𝑑𝑥 𝑑 [𝒇(𝒙)]𝒏 = 𝑛[𝑓(𝑥)]𝑛−1 ∗ 𝑓′(𝑥) 𝑑𝑥 Trig Derivatives: 𝑑 𝑑 𝐬𝐢𝐧(𝒖) = cos(𝑢) 𝑢′ 𝐜𝐨𝐬(𝒖) = − sin(𝑢) 𝑢′ “With Chain Rule” 𝑑𝑥 𝑑𝑥 𝑑 𝑑 𝐭𝐚𝐧(𝒖) = 𝑠𝑒𝑐 2 (𝑢) 𝑢′ 𝐜𝐨𝐭(𝒖) = −𝑐𝑠𝑐 2 (𝑢) 𝑢′ 𝑑𝑥 𝑑𝑥 𝑑 𝑑 𝐬𝐞𝐜(𝒖) = sec(𝑢) tan(𝑢) 𝑢′ 𝐜𝐬𝐜(𝒖) = − csc(𝑢) cot(𝑢) 𝑢′ 𝑑𝑥 𝑑𝑥 Inverse Trig Derivatives: 𝑑 𝑢′ 𝑑 −𝑢′ [𝒔𝒊𝒏−𝟏 (𝒖)] = [𝒄𝒐𝒔−𝟏 (𝒖)] = “With Chain Rule” 𝑑𝑥 √1 − 𝑢2 𝑑𝑥 √1 − 𝑢2 𝑑 𝑢′ 𝑑 −𝑢′ [𝒕𝒂𝒏−𝟏 (𝒖)] = [𝒄𝒐𝒕−𝟏 (𝒖)] = 𝑑𝑥 1 + 𝑢2 𝑑𝑥 1 + 𝑢2 𝑑 𝑢′ 𝑑 −𝑢′ [𝒔𝒆𝒄−𝟏 (𝒖)] = [𝒄𝒔𝒄−𝟏 (𝒖)] = 𝑑𝑥 |𝑢|√𝑢2 − 1 𝑑𝑥 |𝑢|√𝑢2 − 1 Exponential Derivatives: 𝑑 𝒖 [𝒆 ] = 𝑒 𝑢 ∗ 𝑢′ 𝑑𝑥 𝑑 𝒖 [𝒂 ] = 𝑎𝑢 ∗ 𝑢′ ∗ ln 𝑎 𝑑𝑥 Derivatives of Logs: 𝑑 𝑢′ [𝐥𝐧 𝒖] = 𝑑𝑥 𝑢 𝑑 𝑢′ [𝒍𝒐𝒈𝒂 (𝒖)] = 𝑑𝑥 𝑢 ln 𝑎 www.Video-Tutor.net Logarithmic Differentiation: 𝑑 𝒗 𝑣𝑢′ [𝒖 ] = 𝑢𝑣 [ + 𝑣 ′ ln (𝑢)] 𝑑𝑥 𝑢 Inverse Functions: 𝑑 −𝟏 1 [𝒇 (𝒂)] = 𝒇(𝒃) = 𝒂 𝒇−𝟏 (𝒂) = 𝒃 𝑑𝑥 𝑓′(𝑏) 𝑑 −𝟏 1 [𝒇 (𝒙)] = ′ −1 𝑑𝑥 𝑓 [𝑓 (𝑥)] Limit Definition: 𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝒇′ (𝒙) = lim ℎ→0 ℎ Alternative Definition: 𝑓(𝑥) − 𝑓(𝑎) 𝒇′ (𝒂) = lim 𝑥→𝑎 𝑥−𝑎 www.Video-Tutor.net