What is the derivative of (sin(x)^{1/x}) ?

The derivative of (sin(x)^{1/x}) is sin^{1/x}(x)(-(ln(sin(x))}{x^2}+\frac{cot(x))/x)

What is the first derivative of (sin(x)^{1/x}) ?

The first derivative of (sin(x)^{1/x}) is sin^{1/x}(x)(-(ln(sin(x))}{x^2}+\frac{cot(x))/x)

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\frac{d}{d x} ((sin (x))^{\frac{1}{x}})

sin^{\frac{1}{x}} (x) (- \frac{ln ( sin ( x ) )}{x ^{2}} + \frac{cot ( x )}{x})

Solution steps

\frac{d}{d x} ((sin (x))^{\frac{1}{x}})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(sin (x))^{\frac{1}{x}} = e^{\frac{1}{x} l n (s i n (x))}

= \frac{d}{d x} (e^{\frac{1}{x} l n (s i n (x))})

Apply the chain rule:

e^{\frac{1}{x} l n (s i n (x))} \frac{d}{d x} (\frac{1}{x} ln (sin (x)))

= e^{\frac{1}{x} l n (s i n (x))} \frac{d}{d x} (\frac{1}{x} ln (sin (x)))

\frac{d}{d x} (\frac{1}{x} ln (sin (x))) = - \frac{ln ( sin ( x ) )}{x ^{2}} + \frac{cot ( x )}{x}

= e^{\frac{1}{x} l n (s i n (x))} (- \frac{ln ( sin ( x ) )}{x ^{2}} + \frac{cot ( x )}{x})

e^{\frac{1}{x} l n (s i n (x))} = sin^{\frac{1}{x}} (x)

= sin^{\frac{1}{x}} (x) (- \frac{ln ( sin ( x ) )}{x ^{2}} + \frac{cot ( x )}{x})

What is the derivative of (sin(x)^{1/x}) ?
The derivative of (sin(x)^{1/x}) is sin^{1/x}(x)(-(ln(sin(x))}{x^2}+\frac{cot(x))/x)
What is the first derivative of (sin(x)^{1/x}) ?
The first derivative of (sin(x)^{1/x}) is sin^{1/x}(x)(-(ln(sin(x))}{x^2}+\frac{cot(x))/x)