What is the derivative of (sinh(x^2)^{1/x}) ?

The derivative of (sinh(x^2)^{1/x}) is sinh^{1/x}(x^2)(-(ln(sinh(x^2)))/(x^2)+2coth(x^2))

What is the first derivative of (sinh(x^2)^{1/x}) ?

The first derivative of (sinh(x^2)^{1/x}) is sinh^{1/x}(x^2)(-(ln(sinh(x^2)))/(x^2)+2coth(x^2))

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\frac{d}{d x} ((sinh (x^{2}))^{\frac{1}{x}})

sinh^{\frac{1}{x}} (x^{2}) (- \frac{ln ( sinh ( x ^{2} ) )}{x ^{2}} + 2 coth (x^{2}))

Solution steps

\frac{d}{d x} ((sinh (x^{2}))^{\frac{1}{x}})

Apply exponent rule:

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

(sinh (x^{2}))^{\frac{1}{x}} = e^{\frac{1}{x} l n (s i n h (x^{2}))}

= \frac{d}{d x} (e^{\frac{1}{x} l n (s i n h (x^{2}))})

Apply the chain rule:

e^{\frac{1}{x} l n (s i n h (x^{2}))} \frac{d}{d x} (\frac{1}{x} ln (sinh (x^{2})))

= e^{\frac{1}{x} l n (s i n h (x^{2}))} \frac{d}{d x} (\frac{1}{x} ln (sinh (x^{2})))

\frac{d}{d x} (\frac{1}{x} ln (sinh (x^{2}))) = - \frac{ln ( sinh ( x ^{2} ) )}{x ^{2}} + 2 coth (x^{2})

= e^{\frac{1}{x} l n (s i n h (x^{2}))} (- \frac{ln ( sinh ( x ^{2} ) )}{x ^{2}} + 2 coth (x^{2}))

e^{\frac{1}{x} l n (s i n h (x^{2}))} = sinh^{\frac{1}{x}} (x^{2})

= sinh^{\frac{1}{x}} (x^{2}) (- \frac{ln ( sinh ( x ^{2} ) )}{x ^{2}} + 2 coth (x^{2}))

\frac{\partial}{\partial y} (5 x^{4} y)

\frac{d}{d x} (5^{x} - 1)

\frac{sin ( x ) + cos ( x )}{e ^{x}}

\int (3 x + 1) (3 x^{2} + 2 x)^{\frac{1}{3}} d x

\frac{d}{d x} (\frac{1}{x ^{2} - 2 x + 2})

What is the derivative of (sinh(x^2)^{1/x}) ?
The derivative of (sinh(x^2)^{1/x}) is sinh^{1/x}(x^2)(-(ln(sinh(x^2)))/(x^2)+2coth(x^2))
What is the first derivative of (sinh(x^2)^{1/x}) ?
The first derivative of (sinh(x^2)^{1/x}) is sinh^{1/x}(x^2)(-(ln(sinh(x^2)))/(x^2)+2coth(x^2))