. h h [ x x ln x + x ] h ln h + h t ln h + h t e t h + h e t h h e t he t + h h e e t + he t h et + e t e t h,k h k k h et + + e t 4 et + e t 4 k et e

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1 .. x + y. { x cost y sint x, y t x y. { x sect y tant t t t t t cost,sint t r A r t.3 t A.4 t x t x Ph,k x y Ph,k P x M OP M t t t h hk x x t O M

2 . h h [ x x ln x + x ] h ln h + h t ln h + h t e t h + h e t h h e t he t + h h e e t + he t h et + e t e t h,k h k k h et + + e t 4 et + e t 4 k et e t h,k x, y x et + e t y et e t.5 t cost sint t 雙曲函數 hyperbolic function cost sint hyperbolic sine hyperbolic cosine sin,cos h sinhx ex e x coshx ex + e x sine sincosine cos

3 . tanhx sinhx coshx ex e x e x + e x ex e x + cothx coshx sinhx ex + e x e x e x ex + e x sechx coshx e x + e x cschx sinhx e x e x sinhx sinh x e x e x ex e x sinhx coshx y coshx y tanhx y sinhx. x cost, y sint x + y cos t + sin t cos t + tan t sec t sin t cot t + csc t x cosht, y sinht x y cosh t sinh t cosh t tanh t sech t sinh t coth t csch t 3

4 . sinhx + y sinhxcoshy + coshxsinhy coshx + y coshxcoshy + sinhxsinhy tanhx + tany tanhx + y + tanhx tanhy y x sinhx sinhx coshx coshx cosh x + sinh x cosh x + sinh x tanhx tanhx + tanh x x x sinh x cosh x coshx coshx + sin cos x + sin x sinx cosh x sinh x cosx + y cosxcosy sinxsiny sinxsiny coshx + y coshxcoshy + sinhxsinhy 4

5 . sin x cosx sin x sinh x coshx e i x cosx + i sinx i x e x x x e i x + i x x i x3 + 3! x! + + i x x3 3! + cosx + i sinx i e i x cosx i sinx e i x cosx + i sinx e i x cosx i sinx cosx ei x + e i x sinx ei x e i x i cosx ei x + e i x coshi x sinx ei x e i x i sinhi x i i x.6 x coshx cos i x cosi x.6 i sinhx sin i x i sinhx i sin i x i sini x 5

6 . sin sinx siny i sinhi x i sinhi y sinhi xsinhi y i cos coshx + y cosi x + i y cosi xcosi y sini xsini y cosh x cosh y i sinh x i sinh y coshx coshy + sinhx sinhy sin De Moive cosx ± i sinx n cosnx ± i sinnx coshi x ± sinhi x n cosinx ± sininx i x x coshx ± sinhx n coshnx ± sinhnx.3 x sinhx e x e x ex + e x coshx x x coshx e x + e x x tanhx ex e x sinhx x tanhx sinhx x coshx coshxcoshx sinhxsinhx cos x cosh t sinh t cos x sech x 3 x cothx csch x sechx tanhxsechx x cschx cothxcschx x 3 6

7 ..4 y sinh x sinhy x x e y e y e y e y e y xe y 0 e y x ± 4x + 4 x ± x + y sinh x ln x x + + e y sinhy + coshy e y e y + e y + e y sinhy x, coshx sinh y + x + e y sinhy + coshy x + y ln x x + + x + y cosh x e y sinhy + coshy x + x y ln x + x, x 7

8 . tanh x x tanhy ey e y + xe y + x e y e y + x x + x y ln x y tanh x + x ln, x < x x tanh x coth x coth x + x ln, x > x sech x x ln + x x, 0 < x csch + x x ln + x x, x 0 tanhx y < tanh x x <.5 x sinh x x ln x x + + x + x + x + x + x + x x + x + x x x + x + x x + x x + + x x + y x x y x x + 8

9 . y sinh x x sinhy y coshy y x + sinh y + x x sinh x x cosh x x tanh x x coth x x sech x x csch x y sinhy y coshy y tanhy y cothy y sechy y cschy coshy + x sinhy x sech y x csch y x sechytanhy x x cschycothy x + x.6 sinhx ex e x [ + x + x x + x3 3! + x + x3 3! +! + x3 3! + x + x! x3 3! + ] 9

10 . coshx ex + e x [ + x + x + x! + + x! +! + x3 3! + + x + x! x3 3! + ] sinhx e x coshx e x e x sinhx + coshx cothx tanh x x x + x + x 4 + x 6 + C x 0 tanh x C + x + x3 3 + x5 5 + tanh 0 0 C C 0 sinh x + x + x C + x sinh x C +C 0 0 x +C +C x +C x4 +C x 3 3 +C x 5 5 +C3 3 x6 + x C x 0 sinh 0 0 C C 0.7 inversive geometry 0

11 . x + x x tant, x sec t t sec t t sect sect t ln sect + tant +C ln + x + x +C x sinht, x cosht t cosht t t cosht t +C sinh x +C + x x coshtcosht t + cosht t [ t + sinht ] +C [ ] t + sinhtcosht +C [ ] sinh x + x + x +C [ ] ln x + + x + x + x +C

3.2 導函數其切線 (tangent line) 為通過 P, 且其斜率為 m 的直線, 即 y = f(a) + m(x a) (3) 其法線 (normal line) 為通過 P 且與切線垂直的直線, 即 y = f(a) 1 (x a) m

3.2 導函數其切線 (tangent line) 為通過 P, 且其斜率為 m 的直線, 即 y = f(a) + m(x a) (3) 其法線 (normal line) 為通過 P 且與切線垂直的直線, 即 y = f(a) 1 (x a) m 第 3 章微分 (Differentiation) 目錄 3.1 切線................................... 25 3.2 導函數.................................. 26 3.3 微分公式................................. 28 3.4 連鎖律..................................