What is taylor (cos(sqrt(x)))^{1/x} ?

The answer to taylor (cos(sqrt(x)))^{1/x} is

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taylor (cos(sqrt(x)))^{1/x}

taylor

(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

Solution steps

Apply the Taylor Formula

= 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

Evaluate Derivatives

= 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

Refine

= 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

x = y^{2} - 4 y, x = 0

y^{^{'}} + (tan (t)) y = tan (t)

y = (x^{2} + 3 x)^{2}

\frac{d}{d x} (coth (x))

x \to 0 + lim (e^{\frac{1}{x}} \cdot x)

What is taylor (cos(sqrt(x)))^{1/x} ?
The answer to taylor (cos(sqrt(x)))^{1/x} is