What is the derivative of 1/(sqrt(1+sin^2(x))) ?

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

What is the first derivative of 1/(sqrt(1+sin^2(x))) ?

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

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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

f (x) = 2 x^{- \frac{3}{4}}

3 x - 5 x^{2}

\frac{d}{d x} (e^{(3 x^{2} y + 4 z)} - x y^{3} z^{4} + \frac{x ^{2}}{2 y} - 4 x^{3} z^{4})

\int e^{- s t} (1) d t

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