テイラー (cos(sqrt(x)))^{1/x}

解

テイラー

(cos (x))^{\frac{1}{x}}

cos (1) + \frac{cos ( 1 ) ( - 2 ln ( cos ( 1 ) ) - tan ( 1 ) )}{2} (x - 1) + \frac{- sec ^{2} ( 1 ) cos ( 1 ) + tan ^{2} ( 1 ) cos ( 1 ) + 8 cos ( 1 ) ln ( cos ( 1 ) ) + 4 cos ( 1 ) ln ^{2} ( cos ( 1 ) ) + 4 tan ( 1 ) cos ( 1 ) ln ( cos ( 1 ) ) + 5 tan ( 1 ) cos ( 1 )}{8} (x - 1)^{2} + \dots

解答ステップ

テイラーの公式を適用する

= cos (1) + \frac{\frac{d}{d x} ( ( cos ( x ) ) ^{\frac{1}{x}} ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( ( cos ( x ) ) ^{\frac{1}{x}} ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( cos ( x ) ) ^{\frac{1}{x}} ) ( 1 )}{3 !} (x - 1)^{3} + \dots

導関数を評価する

= cos (1) + \frac{\frac{c o s ( 1 ) ( - 2 l n ( c o s ( 1 ) ) - t a n ( 1 ) )}{2}}{1 !} (x - 1) + \frac{\frac{- s e c ^{2} ( 1 ) c o s ( 1 ) + t a n ^{2} ( 1 ) c o s ( 1 ) + 8 c o s ( 1 ) l n ( c o s ( 1 ) ) + 4 c o s ( 1 ) l n ^{2} ( c o s ( 1 ) ) + 4 t a n ( 1 ) c o s ( 1 ) l n ( c o s ( 1 ) ) + 5 t a n ( 1 ) c o s ( 1 )}{4}}{2 !} (x - 1)^{2} + \dots

改良

= cos (1) + \frac{cos ( 1 ) ( - 2 ln ( cos ( 1 ) ) - tan ( 1 ) )}{2} (x - 1) + \frac{- sec ^{2} ( 1 ) cos ( 1 ) + tan ^{2} ( 1 ) cos ( 1 ) + 8 cos ( 1 ) ln ( cos ( 1 ) ) + 4 cos ( 1 ) ln ^{2} ( cos ( 1 ) ) + 4 tan ( 1 ) cos ( 1 ) ln ( cos ( 1 ) ) + 5 tan ( 1 ) cos ( 1 )}{8} (x - 1)^{2} + \dots