What is the derivative of 1/(sqrt(sin(x))) ?

The derivative of 1/(sqrt(sin(x))) is -(cos(x))/(2sin^{3/2)(x)}

What is the first derivative of 1/(sqrt(sin(x))) ?

The first derivative of 1/(sqrt(sin(x))) is -(cos(x))/(2sin^{3/2)(x)}

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

- \frac{cos ( x )}{2 sin ^{\frac{3}{2}} ( x )}

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{2 ( sin ( x ) ) ^{\frac{3}{2}}} \frac{d}{d x} (sin (x))

= - \frac{1}{2 ( sin ( x ) ) ^{\frac{3}{2}}} \frac{d}{d x} (sin (x))

\frac{d}{d x} (sin (x)) = cos (x)

= - \frac{1}{2 ( sin ( x ) ) ^{\frac{3}{2}}} cos (x)

Simplify

- \frac{1}{2 sin ^{\frac{3}{2}} ( x )} cos (x) : - \frac{cos ( x )}{2 sin ^{\frac{3}{2}} ( x )}

= - \frac{cos ( x )}{2 sin ^{\frac{3}{2}} ( x )}

What is the derivative of 1/(sqrt(sin(x))) ?
The derivative of 1/(sqrt(sin(x))) is -(cos(x))/(2sin^{3/2)(x)}
What is the first derivative of 1/(sqrt(sin(x))) ?
The first derivative of 1/(sqrt(sin(x))) is -(cos(x))/(2sin^{3/2)(x)}